Gaussian processes (GPs) are flexible priors for modeling functions. However, their success depends on the kernel accurately reflecting the properties of the data. One of the appeals of the GP framework is that the marginal likelihood of the kernel hyperparameters is often available in closed form, enabling optimization and sampling procedures to fit these hyperparameters to data. Unfortunately, point-wise evaluation of the marginal likelihood is expensive due to the need to solve a linear system; searching or sampling the space of hyperparameters thus often dominates the practical cost of using GPs. We introduce an approach to the identification of kernel hyperparameters in GP regression and related problems that sidesteps the need for costly marginal likelihoods. Our strategy is to "amortize" inference over hyperparameters by training a single neural network, which consumes a set of regression data and produces an estimate of the kernel function, useful across different tasks. To accommodate the varying dimension and cardinality of different regression problems, we use a hierarchical self-attention-based neural network that produces estimates of the hyperparameters which are invariant to the order of the input data points and data dimensions. We show that a single neural model trained on synthetic data is able to generalize directly to several different unseen real-world GP use cases. Our experiments demonstrate that the estimated hyperparameters are comparable in quality to those from the conventional model selection procedures, while being much faster to obtain, significantly accelerating GP regression and its related applications such as Bayesian optimization and Bayesian quadrature. The code and pre-trained model are available at https://github.com/PrincetonLIPS/AHGP.