Let D be a domain in \({\mathcal R}^ m\) (m\(\geq 2)\) with connected boundary and let E be a compact subset of D. It is shown that if u is real-valued, continuous and subharmonic in D, then u can be uniformly approximated in E by subharmonic polynomials. This result with ''harmonic'' in place of ''subharmonic'' is a classical theorem of J. L. Walsh. The paper also contains a criterion, in terms of the associated Riesz measure, for a real-valued subharmonic function to be continuous in a domain.