Optimisation problems pervade structural bioinformatics. In this review, we describe recent work addressing a selection of bioinformatics challenges. We begin with a discussion of research into protein structure comparison, and highlight the utility of Kolmogorov complexity as a measure of structural similarity. We then turn to research into de novo protein structure prediction, in which structures are generated from first principles. In this endeavour, there is a compromise between the detail of the model and the extent to which the conformational space of the protein can be sampled. We discuss some developments in this area, including off-lattice structure prediction using the great deluge algorithm. One strategy to reduce the size of the search space is to restrict the protein chain to sites on a regular lattice. In this context, we highlight the use of memetic algorithms, which combine genetic algorithms with local optimisation, to the study of simple protein models on the two-dimensional square lattice and the face-centred cubic lattice.