Molecular simulations are starting to achieve sufficient length-and time-scales that allow us to efficiently probe the compositional and topological complexity of cellular membranes. Notably, many key functions related to these membranes encapsulating either the entire cell or its organelles involve significant local membrane curvature. The plasma membrane has specific invaginated signaling platforms, the complex and dynamic topography of the mitochondrial inner membrane is involved in numerous biological processes, and the cells store energy in lipid droplets that bud into the cytosol from the membrane surrounding the endoplasmic reticulum. In these processes, curvature is generated by both lipids and proteins. Moreover, they can both also sense curvature, leading to spatial sorting with functional implications. Adequate modelling and analysis of such non-planar lipid bilayers requires the development of algorithms that take into account the geometry of the membrane. A particularly fascinating property impacted by membrane curvature is the lateral diffusion of membrane components, which has a direct influence on the interpretation of experimental results obtained from techniques such as Fluorescence Correlation Spectroscopy (FCS) and Single Particle Tracking (SPT) that often assume planar geometries. Notably, the effects of the underlying curvature can be falsely interpreted as slowed-down or even anomalous diffusion, calling for further clarification by alternative approaches such as computer simulations.