convexity in metric space is the main topic of discussion in this thesis. to undertake the study we have studied extensively the means introduced by doss and included the results concerning means derived by gahier and murphy. we use this definition of a mean to define a new notion of convexity on a metric space, called b-convexity. b-convexity has been compared with other notions of convexity on a metric space. finally following a construction given by machado, we show that a b-convex metric space, satisfying certain properties, is essentially a convex subset of a normed space and the space is unique.