We introduce space-time white noise and spatially homogeneous Gaussian noise, and define stochastic integrals with respect to these noises. We present their fundamental properties and relations with other stochastic integrals in the literature. These are used to define solutions to linear and non-linear stochastic PDEs, including the stochastic heat, wave, Poisson and related equations in various spatial dimensions. In the nonlinear case, we discuss first the case of globally Lipschitz coefficients, then some extensions to locally Lipschitz coefficients with linear growth and some examples with super-linear growth. These lectures assume prior knowledge of stochastic calculus with respect to Brownian motion and continuous martingales, but no particular familiarity with space-time stochastic integrals; they are based on a book with Marta Sanz-Solé that is in preparation.