The study of neurodegenerative diseases relies on the reconstruction and analysis of the brain cortex from magnetic resonance imaging (MRI). Traditional frameworks for this task like FreeSurfer demand lengthy runtimes, while its accelerated variant FastSurfer still relies on a voxel-wise segmentation which is limited by its resolution to capture narrow continuous objects as cortical surfaces. Having these limitations in mind, we propose DeepCSR, a 3D deep learning framework for cortical surface reconstruction from MRI. Towards this end, we train a neural network model with hypercolumn features to predict implicit surface representations for points in a brain template space. After training, the cortical surface at a desired level of detail is obtained by evaluating surface representations at specific coordinates, and subsequently applying a topology correction algorithm and an isosurface extraction method. Thanks to the continuous nature of this approach and the efficacy of its hypercolumn features scheme, DeepCSR efficiently reconstructs cortical surfaces at high resolution capturing fine details in the cortical folding. Moreover, DeepCSR is as accurate, more precise, and faster than the widely used FreeSurfer toolbox and its deep learning powered variant FastSurfer on reconstructing cortical surfaces from MRI which should facilitate large-scale medical studies and new healthcare applications.