The text centres around notions of Frobenius structure which in recent years have drawn some attention in topology, physics, algebra, and computer science. In topology the structure arises in the category of 2-dimensional oriented cobordisms (and their linear representations, which are 2-dimensional topological quantum field theories) – this is the subject of the first chapter. The main result here (due to Abrams [1]) is a description in terms of generators and relations of the monoidal category 2Cob. In algebra, the structure manifests itself simply as Frobenius algebras, which are treated carefully in Chapter 2. The main result here is a characterisation of Frobenius algebras in terms of comultiplication which goes back to Lawvere [32] and was rediscovered by Quinn [43] and Abrams [1]. The main result of these notes is that these two categories are equivalent: the category of 2-dimensional topological quantum field theories and the category of commutative Frobenius algebras. This result is due to Dijkgraaf [16], further details of the proof having been provided by Quinn [43], Dubrovin [19], and Abrams [1]. The notions from category theory needed in order to express this rigorously (monoidal categories and their linear representations) are developed from an elementary level in Chapter 3. The categorical viewpoint allows us to extract the essence of what is going on in the first two chapters, and prove a natural generalisation of the theorem. To arrive at this insight, we carefully review the classical fact that the simplex category Δ is the free monoidal category on a monoid. (This means in particular that there is an equivalence of categories between the category of algebras and the category of ‘linear representations’ of Δ.)