The problem of a few interacting fermions in quantum physics has sparked intense interest, particularly in recent years owing to connections with the behaviour of superconductors, fermionic superfluids and finite nuclei. This review addresses recent developments in the theoretical description of four fermions having finite-range interactions, stressing insights that have emerged from a hyperspherical coordinate perspective. The subject is complicated, so we have included many detailed formulae that will hopefully make these methods accessible to others interested in using them. The universality regime, where the dominant length scale in the problem is the two-body scattering length, is particularly stressed, including its implications for the famous BCS-BEC crossover problem. Derivations and relevant formulae are also included for the calculation of challenging few-body processes such as recombination.