Strain can have a large influence on the properties of materials at the nanoscale. The effect of lattice strain on semiconductor devices has been widely studied, but its influence on colloidal semiconductor nanocrystals is still poorly understood. Here we show that the epitaxial deposition of a compressive shell (ZnS, ZnSe, ZnTe, CdS or CdSe) onto a soft nanocrystalline core (CdTe) to form a lattice-mismatched quantum dot can dramatically change the conduction and valence band energies of both the core and the shell. In particular, standard type-I quantum-dot behaviour is replaced by type-II behaviour, which is characterized by spatial separation of electrons and holes, extended excited-state lifetimes and giant spectral shifts. Moreover, the strain induced by the lattice mismatch can be used to tune the light emission--which displays narrow linewidths and high quantum yields--across the visible and near-infrared part of the spectrum (500-1,050 nm). Lattice-mismatched core-shell quantum dots are expected to have applications in solar energy conversion, multicolour biomedical imaging and super-resolution optical microscopy.