A consensus problem consists of a group of dynamic agents who seek to agree upon certain quantities of interest. The agents exchange information according to a communication network modeled as a directed time-varying graph and evolve in a convex metric space; a metric space endowed with a convex structure. In this paper we generalize the asymptotic consensus problem to convex metric spaces. Under weak connectivity assumptions, we show that if at each iteration an agent updates its state by choosing a point from a particular subset of the generalized convex hull generated by the agent's current state and the states of its neighbors, then agreement is achieved asymptotically. In addition, we present several examples of convex metric spaces and their corresponding agreement algorithms.