A Chebyshev center of a set $A$ in a metric space $(X,d)$ is a point of $X$ best approximating the set $A$ i.e., it is a point $x_0\in X$ such that $\sup_{y\in A}d(x_0,y)=\inf_{x\in X}\sup_{y\in A}d(x,y)$. We discuss the existence and uniqueness of such points in metric spaces thereby generalizing and extending several known result on the subject.