  We present the Walsh theory of stochastic integrals with respect to
martingale measures, alongside of the Da Prato and Zabczyk theory of stochastic
integrals with respect to Hilbert-space-valued Wiener processes and some other
approaches to stochastic integration, and we explore the links between these
theories. We then show how each theory can be used to study stochastic partial
differential equations, with an emphasis on the stochastic heat and wave
equations driven by spatially homogeneous Gaussian noise that is white in time.
We compare the solutions produced by the different theories.
