Variational Bayesian neural networks (BNN) perform variational inference over weights, but it is difficult to specify meaningful priors and approximating posteriors in a high-dimensional weight space. We introduce functional variational Bayesian neural networks (fBNNs), which maximize an Evidence Lower BOund (ELBO) defined directly on stochastic processes, i.e. distributions over functions. We prove that the KL divergence between stochastic processes is equal to the supremum of marginal KL divergences over all finite sets of inputs. Based on this, we introduce a practical training objective which approximates the functional ELBO using finite measurement sets and the spectral Stein gradient estimator. With fBNNs, we can specify priors which entail rich structure, including Gaussian processes and implicit stochastic processes. Empirically, we find that fBNNs extrapolate well using various structured priors, provide reliable uncertainty estimates, and can scale to large datasets.