Conventional (equatorial) Mercator map arbitrarily clipped between parallels 85°N and 85°S 
The great Flemish cartographer Gerhard Kremer became famous with the Latinized name Gerardus Mercator. A revolutionary invention, the cylindrical projection bearing his name has a remarkable property: any straight line between two points is a loxodrome, or line of constant course on the sphere. In the common equatorial aspect, the Mercator loxodrome bears the same angle from all meridians. In other words, if one draws a straight line connecting a journey's starting and ending points on a Mercator map, that line's slope yields the journey direction, and keeping a constant bearing is enough to get to one's destination.
This projection is almost always presented in a tangent case, with the Equator as a standard parallel free of distortion. When using a secant case, two parallels symmetrically opposite the Equator become standard lines; the resulting map is nearly identical after a change in aspect ratio, much like variations of the equalarea cylindrical projection. In this context, the words "tangent" and "secant" are only conceptual, since the Mercator projection is not defined by a perspective process on a developing cylinder.
The only conformal cylindrical projection, Mercator's device was a boon to navigators from the 16thcentury until the present, despite suffering from extreme area distortion near the poles: in order to keep shapes undistorted, Antarctica is enormously stretched, and Greenland is rendered about nine times larger than actual size. Indeed, stretching grows steadily towards the top and bottom of the map (in the equatorial form, in higher latitudes; the poles would be actually placed infinitely far away). Mercator maps seldom extend above the 80°N parallel or below 75°S. Both this apparent shift of the Equator southwards and the areal exaggeration of intermediate latitudes, which mostly coincide with developed nations, have repeatedly incited disapproval about its supposed bias against the Third World (it was even claimed to aid racial discrimination by promoting a supposed superiority of Europe, the U.S.S.R. or the United States); misguided or naïve controversies and proposals to fix the wrong problem include the "Peters" projection, which (like many others) preserves areas, but strongly distorts shapes and has no especially interesting property to compensate for. Although historically several maps have been enlisted in political propaganda, this is no fault of the projections themselvels: like all conformal projections, Mercator's was never intended for world wall maps. Nevertheless, it was once common in textbooks. More recently, the spherical case was chosen for the world view of Google Maps, clipped between the 85°3'4" latitudes, which yield a square map, convenient for efficient storage and retrieval.
Although fundamental, a Mercator map is not the only one used by navigators, as the loxodrome does not usually coincides with the geodesic, except in short travels. The geodesic may be plotted on a gnomonic map, and later transferred to a Mercator map and split in loxodromes piecewise.
This projection was possibly first used by Etzlaub ca. 1511; however, it was for sure only widely known after Mercator's atlas of 1569. Since a rigorous underlying mathematics was not available at the time, Mercator probably defined the graticule by geometric construction; E.Wright formally presented equations in 1599.
More commonly applied to largescale maps, the transverse aspect preserves every property of Mercator's projection, but since meridians are not straight lines, it is better suited for largescale topographic maps than navigation. It does map a single central meridian (two when secant) with no distortion.
Equatorial, transverse and oblique maps offer the same distortion pattern.
The transverse aspect, with equations for the spherical case, was presented by Lambert in his seminal paper (1772). The ellipsoidal case was developed, among others, by the great mathematician Carl Gauss (ca. 1822) and by Louis Krüger (ca. 1912); it is frequently called the Gauss conformal or GaussKrüger projection.
The best known use of the transverse Mercator projection is the specialized form called Universal Transverse Mercator (UTM) projection system.
The UTM defines a grid covering the world between parallels 84°N and 80°S. The grid is divided in sixty narrow zones, each centered on a meridian. Zones are identified by consecutive numbers, increasing from west to east (the first zone, immediately east of the 180° meridian, is numbered 1; zone 31 lies just east of the Greenwich meridian). A set of parallels divides the grid in rows, labeled by letters from C to X (I and O are not used) starting south. Therefore each zone comprises 20 quadrangles, identified by a numberletter pair. Quadrangles are in turn further subdivided in squares 100 kmwide, identified by double letter combinations.
Although no Mercator map is created by a perspective process, the cylinder is a useful visualization aid. The blue strip is zone 13 of the UTM grid; it is part of a cylindrical slice, approximating a spherical lune 6° wide at the equator and clipped by the 84°N and 80°S parallels. This is not the shape of UTM maps; rather, each UTM zone is covered by many partly juxtaposed rectangular largescale sheets. 
Each zone is separately projected using the ellipsoidal form of the transverse Mercator projection with a secant case: scale of the central meridian is reduced by 0.04%, so two lines about 1°37" east and west of it have true scale. The UTM grid was designed for largescale topographic mapping in separate sheets, not for world or regional maps. In particular, sheets from different zones don't juxtapose exactly.
Within each quadrangle, any point may be located by two distances in meters: the easting, east from the central meridian and the northing, north from the Equator. The central meridian's coordinate is always 500,000; the Equator's coordinate is designated 0 for quadrangles in the northern hemisphere, and 10,000,000 for quadrangles in the southern hemisphere. Since the distance from poles to Equator is approximately 10,000 km, such offset origins ensure coordinates (the false eastings and false northings) are always positive.
The UTM grid is fairly regular, with a few exceptions:
The original UTM system was adopted by the U.S. Army in 1949, and variations afterwards by several agencies throughout the world. Despite the name, it is not actually "universal" in the sense that each grid may be based on a different datum, so sheets from different grid sets may or may not be compatible. UTM maps are of course conformal, and distance and area distortion are limited by the large scale of individual sheets.

The azimuthal equidistant projection is trivially easy to draw in the polar aspect and, like all azimuthal designs, it features some special properties for the central point alone: all straight lines touching it are geodesics, and the angle between any two of those lines is the same as on Earth. Hence, oblique azimuthal equidistant maps must be tailormade for each specific location.
The projection's modern name is due to Antonio Cagnoli, who reinvented it in 1799; earlier it had been mentioned by J.Lambert in his seminal paper of 1772, Guillaume Postel (1581), who is often credited as the original author, and Glareanus (ca. 1510), among others.
As a rule, azimuthal projections make straightforward finding true directions from a single point (the reference straight line is, of course, the local meridian): a short wave radio operator whose hardware is stationed at the map's center can use it in order to orient its antenna for maximum gain towards anywhere on Earth.
Furthermore, since of all azimuthal projections the equidistant alone preserves radial distances from the central point, the operator may estimate how much power is required for ensuring stable communication. Likewise, the captain of a nuclear submarine could use this projection to check which cities lie inside its destructive range. Other projections simply are not appropriate.