Virtual knot theory, introduced by L. Kauffman, is a generalization of classical knot theory. It naturally yields the notion of a virtual braid, which is closely related to the notion of a welded braid due to R. Fenn, R. Rimányi and C. Rourke. In this paper we prove that any virtual link or welded link can be described as the closure of a virtual braid or welded braid, respectively, which is unique up to certain basic moves. This is analogous to the Alexander and Markov theorems for classical braids and links.