  This survey article is devoted to the notions of purity, algebraic and
$\Sigma$-algebraic compactness, direct sum decompositions, and representation
type in the category of modules over a ring. It begins with basic definitions,
a brief history, and a discussion of global decomposition problems going back
to work of K\"othe and Cohen-Kaplansky; these are strongly tied to algebraic
compactness. Characterisations of ($\Sigma$-)algebraically compact modules are
presented, as well as the functorial underpinnings on which they are based. In
particular, product-compatible functors, matrix functors and $p$-functors are
discussed. The latter part of the article is devoted to rings of vanishing left
pure global dimension, product completeness, endofiniteness, the pure
semisimplicity problem and its connection with a strong Artin problem on
division ring extensions.
