Classical mereology is based on the idea that parthood is a reflexive, transitive, antisymmetric relation—a weak partial order. Proper parthood is irreflexive, transitive, and asymmetric—serving as the corresponding strict partial order. This chapter critically and carefully evaluates each of these conditions, and considers a range of alternate definitions for proper parthood and other basic notions.. Along the way, it discusses formal relationships between these definitions, and the logical implications of adopting them. The philosophical controversies surrounding these axioms are addressed: discussing self-parts, parthood loops, mutual parts and extensionality, and non-transitive notions such as immediate parthood, local transitivity, and more.