Given a (commutative) ring (with 1) R, the prime spectrum Spec R is the set of prime ideals of R, endowed with the Zariski topolgy: the topology which, in the case that R is a finitely-generated algebra over a field k, has as closed sets the algebraic sets in k^n, i.e. sets which arise as the common zeros of some collection of polynomials. In this talk I will introduce this setting, describe how (and why?) we generalise this to any ring R, and characterise the spaces which arise as Spec R for some ring R (the _spectral_ spaces). I will then discuss how studying spectral spaces gives us insight into prime ideal structure in rings, and give an application to the Kaplansky question (per Discrete Seminar 12th Nov) of characterising those partially-ordered sets which arise as the ordering of prime ideals in some ring.