This paper addresses in an integrated and systematic fashion the relatively overlooked but increasingly important issue of measuring and characterizing the geometrical properties of nerve cells and structures, an area often called neuromorphology. After discussing the main motivation for such an endeavour, a comprehensive mathematical framework for characterizing neural shapes, capable of expressing variations over time, is presented and used to underline the main issues in neuromorphology. Three particularly powerful and versatile families of neuromorphological approaches, including differential measures, symmetry axes/skeletons, and complexity, are presented and their respective potentials for applications in neuroscience are identified. Examples of applications of such measures are provided based on experimental investigations related to automated dendrogram extraction, mental retardation characterization, and axon growth analysis.