Fragment-based drug discovery (FBDD) represents a change in strategy from the screening of molecules with higher molecular weights and physical properties more akin to fully drug-like compounds, to the screening of smaller, less complex molecules. This is because it has been recognised that fragment hit molecules can be efficiently grown and optimised into leads, particularly after the binding mode to the target protein has been first determined by 3D structural elucidation, e.g. by NMR or X-ray crystallography. Several studies have shown that medicinal chemistry optimisation of an already drug-like hit or lead compound can result in a final compound with too high molecular weight and lipophilicity. The evolution of a lower molecular weight fragment hit therefore represents an attractive alternative approach to optimisation as it allows better control of compound properties. Computational chemistry can play an important role both prior to a fragment screen, in producing a target focussed fragment library, and post-screening in the evolution of a drug-like molecule from a fragment hit, both with and without the available fragment-target co-complex structure. We will review many of the current developments in the area and illustrate with some recent examples from successful FBDD discovery projects that we have conducted.