Recently, Convolutional Neural Networks have shown promising results for 3D geometry prediction. They can make predictions from very little input data such as for example a single color image, depth map or a partial 3D volume. A major limitation of such approaches is that they only predict a coarse resolution voxel grid, which does not capture the surface of the objects well. We propose a general framework, called hierarchical surface prediction (HSP), which facilitates prediction of high resolution voxel grids. The main insight is that it is sufficient to predict high resolution voxels around the predicted surfaces. The exterior and interior of the objects can be represented with coarse resolution voxels. This allows us to predict significantly higher resolution voxel grids around the surface, from which triangle meshes can be extracted. Our approach is general and not dependent on a specific input type. In our experiments, we show results for geometry prediction from color images, depth images and shape completion from partial voxel grids. Our analysis shows that the network is able to predict the surface more accurately than a low resolution prediction.