We survey and benchmark traditional and novel learning-based algorithms that address the problem of surface reconstruction from point clouds. Surface reconstruction from point clouds is particularly challenging when applied to real-world acquisitions, due to noise, outliers, non-uniform sampling and missing data. Traditionally, different handcrafted priors of the input points or the output surface have been proposed to make the problem more tractable. However, hyperparameter tuning for adjusting priors to different acquisition defects can be a tedious task. To this end, the deep learning community has recently addressed the surface reconstruction problem. In contrast to traditional approaches, deep surface reconstruction methods can learn priors directly from a training set of point clouds and corresponding true surfaces. In our survey, we detail how different handcrafted and learned priors affect the robustness of methods to defect-laden input and their capability to generate geometric and topologically accurate reconstructions. In our benchmark, we evaluate the reconstructions of several traditional and learning-based methods on the same grounds. We show that learning-based methods can generalize to unseen shape categories, but their training and test sets must share the same point cloud characteristics. We also provide the code and data to compete in our benchmark and to further stimulate the development of learning-based surface reconstruction: https://github.com/raphaelsulzer/dsr-benchmark.