Gaussian processes (GPs) are popular, flexible, and interpretable probabilistic models for functions in geospatial analysis, computer-model emulation, and machine learning. However, direct application of GPs involves dense covariance matrices and is computationally infeasible for large datasets. We consider a framework for fast GP inference based on the so-called Vecchia approximation, which implies a sparse Cholesky factor of the inverse covariance matrix. The approximation can be written in closed form and computed in parallel, and it includes many popular existing approximations as special cases. We discuss applications of the framework to noisy and non-Gaussian data and to the analysis of computer experiments. Further extensions allow nonparametric inference on the covariance matrix, with applications in climate-model emulation and data assimilation.