In the history of mathematics, the algebraic theory of semigroups is a relative new-comer, with the theory proper developing only in the second half of the twentieth century. Before this, however, much groundwork was laid by researchers arriving at the study of semigroups from the directions of both group and ring theory. In this paper, we will trace some major strands in the early development of the algebraic theory of semigroups. We will begin with the aspects of the theory which (...) were directly inspired by, and were analogous to, existing results for both groups and rings, before moving on to consider the first independent theorems on semigroups: theorems with no group or ring analogues. (shrink)