  Since the introduction of Stein's method in the early 1970s, much research
has been done in extending and strengthening it; however, there does not exist
a version of Stein's original method of exchangeable pairs for multivariate
normal approximation. The aim of this article is to fill this void. We present
three abstract normal approximation theorems using exchangeable pairs in
multivariate contexts, one for situations in which the underlying symmetries
are discrete, and real and complex versions of a theorem for situations
involving continuous symmetry groups. Our main applications are proofs of the
approximate normality of rank $k$ projections of Haar measure on the orthogonal
and unitary groups, when $k=o(n)$.
