Abstract: I will present two-sided estimates for the Dirichlet heat kernel on domains that satisfy an inner uniformity condition. This a wide class of domains which can have non-smooth or even fractal boundary, for instance the interior of the Koch snowflake. Gyrya and Saloff-Coste proved two-sided bounds for the Dirichlet heat kernel on unbounded inner uniform domains in a Harnack-type Dirichlet space, i.e. a metric measure space equipped with a strictly local regular Dirichlet form which satisfies a parabolic Harnack inequality. Our estimates extend their result to bounded inner uniform domains. They imply the intrinsic ultracontractivity of the associated semigroup. Moreover, we obtain local estimates for the Dirichlet heat kernel associated with certain non-symmetric forms. For instance, the results apply to any divergence form operator with bounded measurable coefficients and a second order part that is uniformly elliptic and symmetric. This is joint work with Laurent Saloff-Coste.