Kinematic synthesis aims to compute mechanisms that satisfy desired constraints, such as passing through a given point. In exact synthesis, mechanisms must precisely satisfy the constraints and such problems can be considered as applied enumerative geometry such as Alt's problem which asks for the number of four-bar coupler curves which pass through 9 general points in the plane. In approximate synthesis, one has an overdetermined number of constraints so synthesized mechanisms are constructed to minimize an objective function derived from the motion specifications. This talk will demonstrate exact and approximate synthesis for four-bar linkages and highlight techniques from numerical algebraic geometry for solving these problems. This talk will conclude with some open problems in mechanism synthesis and enumerative geometry.