In this paper, we survey recent developments in the birational geometry of the moduli spaces M(ξ) of semistable sheaves on P² following [ABCH], [CH], [CH2], [CHW] and [H]. We discuss the cones of effective, ample and movable divisors on M(ξ). We introduce Bridgeland stability conditions, the main technical tools that precipitated the recent advances, and explain their applications to interpolation problems.