In this memoir we present proofs of basic results, including those developed so far by Harold Bell, for the plane fixed point problem: does every map of a non-separating plane continuum have a fixed point? Some of these results had been announced much earlier by Bell but without accessible proofs. We define the concept of the variation of a map on a simple closed curve and relate it to the index of the map on that curve: Index = Variation + 1. A prime end theory is developed through hyperbolic chords in maximal round balls contained in the complement of a non-separating plane continuum X. We define the concept of an outchannel for a fixed point free map which carries the boundary of X minimally into itself and prove that such a map has a unique outchannel, and that outchannel must have variation -1. Also Bell's Linchpin Theorem for a foliation of a simply connected domain, by closed convex subsets, is extended to arbitrary domains in the sphere. We introduce the notion of an oriented map of the plane and show that the perfect oriented maps of the plane coincide with confluent (that is composition of monotone and open) perfect maps of the plane. A fixed point theorem for positively oriented, perfect maps of the plane is obtained. This generalizes results announced by Bell in 1982. A continuous map of an interval I ⊂ ℝ to ℝ which sends the endpoints of I in opposite directions has a fixed point. We generalize this to maps on noninvariant continua in the plane under positively oriented maps of the plane (with appropriate boundary conditions). Similar methods imply that in some cases noninvariant continua in the plane are degenerate. This has important applications in complex dynamics. E.g., a special case of our results shows that if X is a nonseparating invariant subcontinuum of the Julia set of a polynomial P containing no fixed Cremer points and exhibiting no local rotation at all fixed points, then X must be a point. It follows that impressions of some external rays to polynomial Julia sets are degenerate. © 2012 American Mathematical Society.