Neural implicit functions have recently shown promising results on surface reconstructions from multiple views. However, current methods still suffer from excessive time complexity and poor robustness when reconstructing unbounded or complex scenes. In this paper, we present RegSDF, which shows that proper point cloud supervisions and geometry regularizations are sufficient to produce high-quality and robust reconstruction results. Specifically, RegSDF takes an additional oriented point cloud as input, and optimizes a signed distance field and a surface light field within a differentiable rendering framework. We also introduce the two critical regularizations for this optimization. The first one is the Hessian regularization that smoothly diffuses the signed distance values to the entire distance field given noisy and incomplete input. And the second one is the minimal surface regularization that compactly interpolates and extrapolates the missing geometry. Extensive experiments are conducted on DTU, BlendedMVS, and Tanks and Temples datasets. Compared with recent neural surface reconstruction approaches, RegSDF is able to reconstruct surfaces with fine details even for open scenes with complex topologies and unstructured camera trajectories.