  We prove existence of a countable family of spherically symmetric
self-similar wave maps from 3+1 Minkowski spacetime into the 3-sphere. These
maps can be viewed as excitations of the ground state wave map found previously
by Shatah. The first excitation is particularly interesting in the context of
the Cauchy problem since it plays the role of a critical solution sitting at
the threshold of singularity formation. We analyze the linear stability of our
wave maps and show that the number of unstable modes about a given map is equal
to its excitation index. Finally, we formulate a condition under which these
results can be generalized to higher dimensions.
