Regression on function spaces is typically limited to models with Gaussian process priors. We introduce the notion of universal functional regression, in which we aim to learn a prior distribution over non-Gaussian function spaces that remains mathematically tractable for functional regression. To do this, we develop Neural Operator Flows (OpFlow), an infinite-dimensional extension of normalizing flows. OpFlow is an invertible operator that maps the (potentially unknown) data function space into a Gaussian process…