The interfaces of polar oxide heterostructures can display electronic properties unique from the oxides they border, as they require screening from either internal or external sources of charge. The screening mechanism depends on a variety of factors, including the band structure at the interface, the presence of point defects or adsorbates, whether or not the oxide is ferroelectric, and whether or not an external field is applied. In this review, we discuss both theoretical and experimental aspects of different screening mechanisms, giving special emphasis to ways in which the mechanism can be altered to provide novel or tunable functionalities. We begin with a theoretical introduction to the problem and highlight recent progress in understanding the impact of point defects on polar interfaces. Different case studies are then discussed, for both the high thickness regime, where interfaces must be screened and each interface can be considered separately, and the low thickness regime, where the degree and nature of screening can be manipulated and the interfaces are close enough to interact. We end with a brief outlook toward new developments in this rapidly progressing field.