The formation of ice in clouds can occur through primary processes, either homogeneously or heterogeneously triggered by aerosol particles called ice nuclei, as well as through secondary processes. The homogeneous ice nucleation process involves only pure water or solution droplets. Homogeneous freezing is crucial for the microphysics in the formation of high-altitude cirrus and polar stratospheric clouds, and also in the glaciation of thunderclouds, at temperatures below about 235 K. Nucleation rates in supercooled water have been measured using different experimental techniques: expansion cloud chambers, water-in-oil emulsions, levitation methods, free falling droplets, supersonic nozzles, field measurements, and molecular dynamics simulations. An important question concerns the possibility that the nucleation process in supercooled water can occur not only in the interior volume of the droplet, but even at or close to its surface. Even if there is no conclusive evidence, the majority of experimental and theoretical results suggest that the contribution of surface nucleation increases with decreasing radius of the supercooled droplets, and the surface (or sub-surface) nucleation rate is prevalent for droplets with radius lower than about 5 μm. If homogeneous freezing initiates at the droplet surface, the freezing rate should depend on the droplet size, and even a slight contamination by molecules within the surface layer could hamper the rate of the nucleation process.