In this course we will first define the concept of mild solution to the stochastic heat equation driven by a space-time white noise by recalling the theory of stochastic integrals with respect to Gaussian random fields. Then, under the assumption that the coefficients are locally Lipschitz functions, we will define and show local existence and uniqueness of the solution. Then, we will prove recent results that give necessary and sufficient conditions on the coefficients for the solution to blow up in finite time, which are related to the well-known Osgood condition for ordinary differential equations.