I) Preventing Threats.

II) Gaining Time.
(In endgame play, accuracy and economy are needed; major attention is therefore must be given to moves that gain or save time. This makes sense in positions where the strategic goal is clear and simple; e.g., queening a passed pawn or obtaining control of a key square. It is logical to expect the King to have a vital role to play in the fight for time.)

III) Preparing For Action
(You want to do something, but the King is either in the way or of no help. Move it to get ready for action!)

IV) Weaving Mating Nets.

V) King Triangulation
(The term "King triangulation" refers to the King making a "triangular" run around its location in order to arrive again at its starting point, but in a position a move behind its opponent. The purpose of triangulation is to give the move to the opponent and is of significance only in zugzwang positions. The greatest effect of successful triangulation occurs in king-and-pawn endgames.)

Read my kibitz at each game for a commentary on the purpose of the King move.