In this paper, we shall discuss convexity and fixed point theorems in certain metric space which are described in an abstract form. At first we shall introduce a concept of convexity in a metric space and study the properties of the space which we call a convex metric space. Futhermore, we formulate some fixed point theorems for nonexpansive mappings (ie mappings which do not increase distances) in the space. Consequently, these generalize fixed point theorems which have been previously proved by Browder [1], Kirk [6] and the author [7] in a Banach space.