Drop impact onto surfaces has long been a popular and important subject of experimental, numerical and theoretical studies to explain phenomena observed both in nature and in many engineering applications. Progress in understanding and describing the hydrodynamics involved in drop impacts has been rapid in recent years, due partly to the availability of high-speed cameras, but also because of accompanying advances in theoretical and numerical approaches. Thus, for simple surfaces, i.e. smooth surfaces of uniform chemistry, the outcome of a drop impact can be well predicted over a large range of impact parameters, including quantitative values of spread dynamics and splash characteristics. This article comprehensively reviews the present level of understanding for such impact situations. However many practical applications involve impacts onto surfaces of higher complexity, either morphologically or chemically, involving textured or porous surfaces or surfaces with non-uniform wettability characteristics. This expands greatly the parameter space for which descriptions of the impact must be found and the present understanding is significantly more rudimentary compared to drop impacts onto simple surfaces. In this review such impacts are discussed by considering effects introduced by morphological changes to the surface and by changes of the wettability. Comparisons to corresponding impacts onto simple surfaces are drawn to underline the additional physical mechanisms that must be considered.