For many mapping applications, like topography and certain kinds of navigation, a lesser constraint, conformality or fidelity of shape, is the most fundamental requisite: at the intersection of any two lines on the map, the angle between is the same as between their counterparts on the sphere; in particular, each parallel must cross every meridian at right angles. Also, at any point the scale distortion, either compression or exaggeration, must be the same in all directions. Conformality is a strictly local property: angles (therefore shapes) are not expected to be preserved much beyond the intersection point; in fact, straight lines on the sphere are usually curved along the plane, and vice versa.
Where conformality fails: singularities 
In nearly every map classified as "conformal",
at least one point is
a singularity, not
actually represented conformally:
Consider the polar conic conformal map above on the left; its central pole is nonconformal owing to the fact that a curve around it (green) completes a 254°33' circuit instead of 360° like in the globe, which is represented by the partially transparent oblique azimuthal orthographic map on the right. The opposite pole (purple) is the second singularity, necessarily absent on the map. Everywhere else, the graticule lines intersect at right angles, a necessary but not sufficient condition for conformality. Notice

Conformal map projections are frequently employed in largescale applications, and seldom used for continental or world maps (those shown here are included for comparison only), although interruption may alleviate this issue. Since no conformal map can be equalarea — most in fact grossly distort dimensions far from the center of the map — conformal projections are almost never applied to thematic and statistical mapping, where comparisons based on size are common.
Albeit one very important conformal projection is one of the oldest map projections still in use, systematic understanding of requisites and properties of conformality had to wait for the development of sophisticated mathematical tools like differential calculus and complex analysis in the 18th and 19th centuries. Conversely, conformal mapping grew into an important branch of modern mathematics. It is also a remarkable engineering tool: for instance, since several flow problems can be more easily solved in simple shapes like a circle or a square, a complex shape can be conformally mapped to the simple shape, the problem solved, then the solution transformed back to the original context via the inverse conformal mapping.
In a short section of his manuscript of 1772 describing two other seminal conformal projections, the conic orthomorphic and the transverse Mercator, Lambert described a relatively simple but very interesting approach:
The result is a class of conformal — because the composition of sucessive conformal mappings is itself conformal— projections with graticules comprising circular arcs; the only straight lines are the central meridian and one base parallel, which always lies midway between the poles (there are two special cases noted below). Scale distortion is large near the poles, which are nonconformal.
Lambert illustrated his paper with a map using a factor 0.5 and the Equator as the straight parallel; this maps the whole Earth into a disc which of course coincides with the inner hemisphere of the original equatorial azimuthal stereographic projection, with total meridian angles at poles 180° instead of 360°. He emphasized the effect of changing the factor and base parallel, but noted other alternatives as inferior, except two special cases: the factor 1 which yields the original stereographic and 0 for the equatorial Mercator.
This class of projections, even the basic case preferred by Lambert, is usually known as the "Lagrange" projection after a famous promoter, the accomplished mathematician Joseph Lagrange who developed its ellipsoidal case and thoroughly studied its properties (1779). For instance, he proposed parameters which minimized the rate of change of scale near any given point, illustrating it with a map for Berlin.
Today, the "Lagrange" projection is almost never used per se, but it has become a fundamental step for the mathematical development of projections, since the sphere conformally mapped to a unit disc is a convenient basis for further transformations. Lambert's method of compressing longitudes was later applied by Aitoff and Hammer on the azimuthal equidistant and azimuthal equalarea projections respectively. A comparison of longitude factors on a finite base projection is instructive and maybe easier to visualize than for the azimuthal stereographic.
Eisenlohr's conformal map 
August's epicycloidal conformal map 
The design published by Friedrich Eisenlohr in 1870 has two additional features: the scale is constant along the boundary meridians; and more remarkably, scale range is the narrowest of any conformal projection, — compare with the range for the three classic projections. Relatively complex calculations initially restricted its use.
The projection designed by Friedrich August and codeveloped by Bellermann was published in 1874 as an alternative to Eisenlohr's design: the range of scale distortion is wider (1 : 8) and not constant at the boundary meridians, but its construction is somewhat simpler. A world map is bounded by an epicycloid, the shape defined by a point on a circle rolling without sliding around another, fixed, circle.
Neither Eisenlohr's nor August's projection should be confused with
similarlooking designs like the generalized "Lagrange" and the
nonconformal
American polyconic,
rectangular
polyconic and
van der Grinten's IV.
Peirce's Quincuncial conformal map in a square 
While working at the U.S. Coast and Geodetic Survey, the American philosopher and polymath Charles Sanders Peirce disclosed his projection in 1879. In the normal aspect, it presents the northern hemisphere in a square; the other hemisphere is split into four triangles symmetrically surrounding the first one, akin to starlike projections. In effect, the whole map is a square, inspiring Peirce to call his projection quincuncial, after the arrangement of five items in a cross.
Tiled Peirce Quincuncial maps 
Peirce's projection is conformal everywhere except at the corners of the inner hemisphere (thus the midpoints of edges in the whole map), where the Equator breaks abruptly. Scale is highly stretched near those four points; conversely, polar regions are rather compressed. The Equator and four meridians are straight but broken lines; all other graticule lines are complex curves.
Pieces of a quincuncial map can evidently be rearranged as a 2:1 rectangle. Also, the map tessellates the plane; i.e., with a trivial rotation, repeated copies can completely cover (tile) an arbitrary area, each copy's features exactly matching those of its neighbors (however, points in the vicinity of the singularities appear twice on the tesselation).
Peirce's projection in a transverse aspect 
Only a few years after Peirce, Émile Guyou from France presented his conformal projection (18861887). In its original form, it comprises the western and eastern hemispheres, each in a square; the Equator and four meridians are straight lines, two of the later broken along the squares' edges. Other meridians and parallels are complex curves.
Guyou map, central meridian 20°E 
Oblique Guyou map, or two Adams hemispheres; central meridian 25°W 
Actually, Peirce's and Guyou's projections are transverse cases of each other, emphasizing polar and equatorial aspects, respectively. Guyou maps can also tile the plane.
Tiled Guyou maps 
Exactly as in the other aspects by Peirce and Guyou, at the square's corners, where conformality fails, scale distortion is extreme. Only the Equator and the central meridian are straight lines; the boundary meridians are also straight but broken at the Equator.
Despite interest due to their mathematical development, the conformal
projections in squares of Peirce, Guyou and Adams were seldom used.
Adams's world in a square (1929) 
Adams's world in a square (1936) 
Although the works of Schwarz and Christoffel realized a constructive proof of Riemann's theorem, their application to cartography (other than simple, particular instances) remained impractical for nearly one century; for most cases, they do not result in closed formulas and require solving a system of nonlinear equations. Actual mapping involves lengthy numerical evaluation by successive approximations.
Even after digital computers became generally available, results were far from uniform. Many algorithms for SchwarzChristoffel mapping suffered from low efficiency, limited precision, or instability, i.e., failure to converge to a result, or poor handling of singularities (usually present at polygon vertices).
After presenting his conformal hemispheres in squares, O.S.Adams proposed two projections with a world map in a single square.
The first one (1929) has poles in opposite corners; scale distortion is extreme at each corner, which lacks conformality. The second version (1936) has poles at midpoints of opposite edges. Again, there's strong scale distortion at the vertices. This projection is not conformal at each corner and the two poles.
Other lessknown conformal projections by Adams were based on an ellipse and several other polygons.
Xarax's world in half of a regular hexagon. Redrawn after Xarax. 
Laurence P. Lee, distinguished cartographer and senior officer at a national mapping agency in New Zealand, further generalized and improved the accuracy of methods for arbitrary conformal mapping. His projections included maps of the world on rectangles, ellipses, the equilateral triangle, the regular tetrahedron (1965) and other regular polyhedra (1976).
Like Adams's, Lee's designs attracted academic interest and paved the way for new mathematical achievements, but found limited usage in common maps.
Constant Xarax from Greece, influenced by Lee's conformal maps on polygons and polyhedra, Briesemeister's oblique projection and polyhedral maps in a butterfly arrangement, proposed a conformal map of the world in half a regular hexagon (2004). Essentially, Lee's tetrahedral conformal projection is drawn in the original south polar aspect, then split into three lobes, which are rearranged around the north pole; the result balances legibility, low interruption count and easily recognizable shapes. Like in the original design, conformality fails at the north pole and at the four points on the edge of the map where the three straight meridians are broken.
Assorted orthographic views of Gilbert's doubleworld projection (conformal on the sphere, not on flat maps).  
An equatorial aspect can present the whole world.  An oblique aspect evidences how every point on Earth is duplicated. 
Conformality is absent at both poles, as shown by a south polar aspect.  DeLucia/Snyder's slightly oblique view. 
Using the same approach of Lambert's compression of the equatorial azimuthal stereographic which originated the "Lagrange" projection, Edgar N. Gilbert (ca. 1970) shrunk longitudes of the sphere itself, with a multiplier 0.5, then moved the latitudes to restore conformality. The whole Earth is remapped into a hemisphere, which can be duplicated in the final "map". Gilbert had at least one instance of this "double globe" actually assembled where each point on Earth is represented twice. The whole sphere is conformal except at either pole.
Seriously?  
Martin Gardner quotes Gilbert as mentioning that few people notice anything odd about his repeated globe. In a lighthearted presentation on unusual maps, Waldo Tobler suggests that hardly anybody objects to an equatorial orthographic aspect of a globe with simple meridian compression, with no conformal correction. Do those reflect some innate tendency to ignore geographic details, a general lack of familiarity with globes, or the authors's facetiousness? 
Somewhat like Raisz's orthoapsidal proposals, Alan DeLucia and John Snyder applied (1986) an orthographic transformation to Gilbert's "twoworld" sphere, creating a flat map centered at 5°N 5°E. As typical of orthographic views, the eastern and western borders of the map are considerably compressed; a thin lune of the Arctic Ocean appears twice and a corresponding slice of Antarctica omitted. Visually the result resembles the globular appearance of the Lagrange and van der Grinten projections, with the polar area exaggeration reminiscent of Mercator maps. Like other orthographic representations of the sphere, the graticule comprises elliptical arcs and the map is neither conformal nor equalarea.