In this chapter we shall discuss the notion of the heat kernel on a metric measure space (M, d, μ). Loosely speaking, a heat kernel pt(x, y) is a family of measurable functions in x, y ∈ M for each t > 0 that is symmetric, Markovian and satisfies the semigroup property and the approximation of identity property. It turns out that the heat kernel coincides with the integral kernel of the heat semigroup associated with the Dirichlet form in L2(M, μ).