Atmospheric ice nucleation is crucial for global precipitation and affects the structure, lifetime and reflectivity of clouds, thereby impacting climate. Ice nucleates in various ways from aerosol particles, termed ice-nucleating particles, over an extensive temperature and humidity range. Quantifying the kinetic and thermodynamic regimes of nucleation is necessary to relate fundamental physics to theoretically based predictions of ice formation for implementation in cloud and climate models. We review how the molecular picture of ice nucleation has advanced in recent years and consequential impacts on the interpretation and parameterization of ice nucleation. Advances include the role of interfacial free energy and pressure on ice nucleation rates, mobility regions of water that generate the critical ice nucleus, classical and non-classical pathways of nucleation, the type of ice polymorph that forms, the impact of solutes on freezing and the role of nanopores as surface features promoting ice nucleation. We also introduce currently debated and evaluated freezing parameterizations for application in model environments. Finally, we outline what we believe are the current needs for improving predictive understanding of ice nucleation. Predicting atmospheric ice formation from aerosol particles for cloud and climate modelling remains challenging. This Review summarizes recent fundamental advances on the governing parameters that lead to ice nucleation from liquid droplets and solid substrates, applying experiments and computational theory.