  In this work, we study the asymptotic behaviour of solutions to the heat
equation in exterior domains, i.e., domains which are the complement of a
smooth compact set in $\mathbb{R}^N$. Different homogeneous boundary conditions
are considered, including Dirichlet, Robin, and Neumann conditions for
integrable initial data in $L^1(\Omega)$. After taking into account the loss of
mass of the solution through the boundary, depending on the boundary
conditions, we describe the asymptotic spatial distribution of the remaining
mass in terms of the Gaussian and of a suitable asymptotic profile function. We
show that our results have optimal time rates.
