Currently, it is hard to reap the benefits of deep learning for Bayesian methods.We present Prior-Data Fitted Networks (PFNs), a method that allows to employ large-scale machine learning techniques to approximate a large set of posteriors.The only requirement for PFNs is the ability to sample from a prior distribution over supervised learning tasks (or functions).The method repeatedly draws a task (or function) from this prior, draws a set of data points and their labels from it, masks one of the labels and learns to make probabilistic predictions for it based on the set-valued input of the rest of the data points.Presented with samples from a new supervised learning task as input, it can then make probabilistic predictions for arbitrary other data points in a single forward propagation, effectively having learned to perform Bayesian inference.We demonstrate that PFNs can near-perfectly mimic Gaussian processes and also enable efficient Bayesian inference for intractable problems, with over 200-fold speedups in multiple setups compared to current methods. We obtain strong results in such diverse areas as Gaussian process regression and Bayesian neural networks, demonstrating the generality of PFNs.