We will give a symplectic geometer`s introduction to hypersurface cusp singularities, i.e. singularities of the form x^p+y^q+z^r+cxyz, for 1/p+1/q+1/r <1 and c a non-zero constant. Loosely speaking, these are the "next most complicated" singularities after the simple (i.e. ADE) singularities. We will focus on properties of the Milnor fibres of these singularities; these are the open complex surfaces obtained by smoothing them. We will first explain how to get explicit descriptions of these as total spaces of Lefschetz fibrations. We will then use these fibrations to: 1. construct some new examples of exact Lagrangian tori in these surfaces, and all Milnor fibres of non-ADE hypersurface singularities. 2. give a proof of homological mirror symmetry for the hypersurface cusp singularities, which ties into Gross, Hacking and Keel`s proof of Looijenga`s conjecture on cusp singularities. No prior knowledge of singularity theory will be assumed.