The main result of this thesis is the two-sided heat kernel estimates for both Dirichlet and Neumann problem in any inner uniform domain of the Euclidean space $\mathbb R^n$. The results of this thesis hold more generally for any inner uniform domain in many other spaces with Gaussian-type heat kernel estimates. We assume that the heat equation is associated with a local divergence form differential operator, or more generally with a strictly local Dirichlet form on a complete locally compact metric space. Other results include the (parabolic) Harnack inequality and the boundary Harnack principle.