The algebraic theory of semigroups emerged in the mid-twentieth century in a form that is sometimes referred to as the ‘classical’ theory, first brought together coherently in the textbooks of Lyapin (Semigroups, 1960) and Clifford and Preston (The algebraic theory of semigroups, 1961–67). In this chapter, we survey aspects of three of the major themes that can be found in that early theory: the study of arbitrary transformations of a set, efforts to mimic familiar results about rings in a semigroup setting, and the questions arising from differential geometry that gave rise to the notion of an inverse semigroup.