Though Fredholm determinants mark the beginning of modern operator theory and have found many applications, e.g., in mathematical physics and random matrix theory, their numerical evaluation has widely been thought to depend on alternative analytic expressions, such as Painlevé representations, or on explicitly known spectra, both available in certain specific cases only. In contrast, in this talk we will show the convergence of general purpose numerical methods, in particular Galerkin schemes for trace class operators and quadrature schemes for integral operators. The convergence rates are exponential if the kernel is holomorphic, which is typically the case for applications in random matrix theory. This new numerical approach was used in a joint work with Ferrari and Prähofer to provide convincing evidence against a conjecture about GOE matrix diffusion. We will demonstrate that the numerical calculations get a near symbolic feel if coded within the chebop system, a development that is joint work with Driscoll and Trefethen.