This paper concerns the Rayleigh--Ritz method for computing an approximation to an eigenpair $(\lambda, x)$ of a non-Hermitian matrix $A$. Given a subspace $\clw$ that contains an approximation to $x$, this method returns an approximation $(\mu, \tilde x)$ to $(\lambda, x)$. We establish four convergence results that hold as the deviation $\epsilon$ of $x$ from $\clw$ approaches zero. First, the Ritz value $\mu$ converges to $\lambda$. Second, if the residual $A\tilde x-\mu\tilde x$ approaches zero, then the Ritz vector $\tilde x$ converges to $x$. Third, we give a condition on the eigenvalues of the Rayleigh quotient from which the Ritz pair is computed that insures convergence of the Ritz vector. Finally, we show that certain unconditionally. (Also cross-referenced as UMIACS-TR-99-08)