We describe a linear-time algorithm for 4-coloring planar graphs. We indeed give an O(V+E+|χ|+1)-time algorithm to C-color V-vertex E-edge graphs embeddable on a 2-manifold M of Euler characteristic χ where C(M) is given by Heawood's corrected (minimax optimal) formula. Finally, there is a linear-time algorithm to 5-color a graph embedded on any fixed surface M except that an M-dependent constant number of vertices are left uncolored. All the algorithms are simple and practical and run on a deterministic pointer machine, except for planar graph 4-coloring which involves enormous constant factors and requires an integer RAM with a random number generator. All of the algorithms mentioned so far are in the ultraparallelizable deterministic computational complexity class "NC." We also have more practical planar 4-coloring algorithms that can run on pointer machines in O(VlogV) randomized time and O(V) space, and a very simple O(V)-time coloring algorithm for planar graphs which conjecturally uses 4 colors.