Large-interaction limits of certain systems of elliptic and parabolic PDE, such as, for instance, population systems with large competition, both provide a powerful mathematical tool that can be exploited to obtain information about systems that are otherwise difficult to analyse, and correspond to important biological and physical phenomena such as spatial segregation, phase separation, or fast chemical reactions. In the case of systems of two equations, a simple but crucial observation that underpins the study of these limits is that a linear combination of the two components typically satisfies a scalar equation with no explicit dependence on the interaction parameter k, the large-interaction limit of which can yield information about the behaviour of the original system when k is large but finite. We will present some of the key ideas of this approach together with applications including an interesting strong-competition limit problem that arises when populations compete strongly on only part of the spatial domain. This talk is mainly based on joint work with Norman Dancer.