Cortical surface reconstruction facilitates both visualisation and quantification of the cerebral cortex, playing a pivotal role in the diagnosis of neurological disorders and the characterisation of cortical folding patterns. Due to the highly folded anatomical structure and intrinsic topological constraints, it is challenging to extract anatomically and topologically accurate cortical surfaces from brain magnetic resonance imaging (MRI) data. In this thesis, we leverage advanced geometric deep learning (DL) techniques and present a series of DL-based frameworks for fast and explicit cortical surface reconstruction from adult, fetal and neonatal brain MRI scans. Firstly, we propose a novel deep neural network architecture for explicit and topology-preserving cortical surface reconstruction end-to-end from adult brain MRI. To prevent surface self-intersections, we further introduce neural ordinary differential equations to learn diffeomorphic surface deformations for cortical surface reconstruction. Secondly, we develop customised DL-based frameworks for fetal and neonatal subjects that undergo rapid brain development. To tackle considerable brain variations across different ages, we utilise an attention mechanism to learn neonatal cortical surface construction conditioned on the ages of neonates. Furthermore, we devise a weakly supervised framework for fetal cortical surface reconstruction supervised by brain segmentations, thereby eliminating the reliance on pseudo ground truth cortical surfaces generated by traditional neuroimage processing pipelines. Finally, we present a fast and robust DL-based pipeline for cortical surface-based structural MRI processing of developing human brains. We demonstrate that the DL-based cortical surface reconstruction approaches proposed in this thesis achieve superior geometrical and topological accuracy, fast inference within only a few seconds, and high adaptability to brain MRI acquired from subjects in various age groups including adults, fetuses and neonates.