We analyze limits and colimits in the category Part of partial groups, algebraic structures introduced by A. Chermak. We will prove that Part is both complete and cocomplete and, in addition, that the full subcategory of finite partial groups is both finitely complete and finitely cocomplete. Cocompleteness is then used in order to define quotients of partial groups. We will also identify a category richer than Set (the category of sets and set-maps) and build the free partial groups over objects is such category; this yields a larger class of free partial groups, eventually allowing to prove that every partial group is the quotient of a free partial group.