This module is an introduction to the calculus side of Singularity Theory. Theory of singularities of differentiable maps is a far-reaching generalisation of the study of functions at maxima and minima. It has numerous applications in mathematics, the natural sciences and technology (as in the so-called theory of bifurcations and catastrophes). This module concentrates on the theory and stability of smooth maps, and classification techniques for critical points of smooth functions. Although not pre-requisites, any of MATH244 (Linear algebra and geometry), MATH343 (Group theory), and MATH349 (Differential geometry) would be helpful. MATH410 (Manifolds, homology and Morse theory) is a follow-up module but may be taken simultaneously.