In this study, we construct a spatio-temporal surface atlas of the developing cerebral cortex, which is an important tool for analysing and understanding normal and abnormal cortical development. In utero Magnetic Resonance Imaging (MRI) of 80 healthy foetuses was performed, with a gestational age range of 21.7 to 38.9 weeks. Topologically correct cortical surface models were extracted from reconstructed 3D MRI volumes. Accurate correspondences were obtained by applying a joint spectral analysis to cortices for sets of subjects close to a specific age. Sulcal alignment was found to be accurate in comparison to spherical demons, a state of the art registration technique for aligning 2D cortical representations (average Fréchet distance ≈ 0.4 mm at 30 weeks). We construct consistent, unbiased average cortical surface templates, for each week of gestation, from age-matched groups of surfaces by applying kernel regression in the spectral domain. These were found to accurately capture the average cortical shape of individuals within the cohort, suggesting a good alignment of cortical geometry. Each spectral embedding and its corresponding cortical surface template provide a dual reference space where cortical geometry is aligned and a vertex-wise morphometric analysis can be undertaken.