A classical result in SCV is the fact that a nonconstant holomorphic map sending a piece of the unit sphere in $\mathbb C^N$ into itself is necessarily locally biholomorphic (and, in fact, extends as an automorphism of the unit ball). Generalizations and variations of this result for mappings between real hypersurfaces have been obtained by a number of mathematicians over the last 30 years. In this talk, we shall discuss some recent joint work with L. Rothschild along these lines for mappings between CR manifolds of higher codimension.