\abstract This paper deals with mathematical issues relating to the computation of spectra of self adjoint operators on Hilbert spaces. We describe a general method for approximating the spectrum of an operator $A$ using the eigenvalues of large finite dimensional truncations of $A$. The results of several papers are summarized which imply that the method is effective in most cases of interest. Special attention is paid to the Schr\"odinger operators of one-dimensional quantum systems. We believe that these results serve to make a broader point, namely that numerical problems involving infinite dimensional operators require a reformulation in terms of \cstar s. Indeed, it is only when the given operator $A$ is viewed as an element of an appropriate \cstar\ $\Cal A$ that one can see the precise nature of the limit of the finite dimensional eigenvalue distributions: the limit is associated with a tracial state on $\Cal A$. For example, in the case where $A$ is the discretized Schr\"odinger operator associated with a one-dimensional quantum system, $\Cal A$ is a simple \cstar\ having a unique tracial state. In these cases there is a precise asymptotic result. \endabstract