This is a report on aspects of the theory and use of monoidal categories. The first section introduces the main concepts through the example of the category of vector spaces. String notation is explained and shown to lead naturally to a link between knot theory and monoidal categories. The second section reviews the light thrown on aspects of representation theory by the machinery of monoidal category theory, such as braidings and convolution. The category theory of Mackey functors is reviewed in the third section. Some recent material and a conjecture concerning monoidal centres is included. The fourth and final section looks at ways in which monoidal categories are, and might, be used for new invariants of low-dimensional manifolds and for the field theory of theoretical physics.

In essence, this paper consists of the notes of four lectures delivered in May 2011 as part of the Chaire de la Vall\'ee Poussin 2011 . The third and fourth lectures were also part of the conference Category Theory, Algebra and Geometry on 26 and 27 May 2011 in Louvain-la-Neuve, Belgium