A metric space \((X,d)\) is strictly convex if for each \(x,y \in X\) and each \(t\in [0,1]\) there is a unique \(z\in X\) so that \(d(x,z)= td(x,y)\), \(d(z,y) (1-t)d(x,y)\). A few results of the following kind are obtained. Let \(K\) be a compact, convex subset of a strictly convex metric space \(X\) with convex round balls and let \(F\) be a commutative family of self-mappings of \(K\) such that each \(f\in F\) belongs to a class \(M\). Then there is a common fixed point for \(F\). Here \(M\) denotes the classes of nonexpansive, quasi-nonexpansive and asymptotically nonexpansive mappings.