The heat kernel plays a central role in mathematics. It occurs in several fields: analysis, geometry and - last but not least - probability theory. In this survey, we shall focus on its analytic aspects, specifically sharp bounds, in the particular setting of Riemannian symmetric spaces of noncompact type. It is a natural tribute to Karpelevic, whose pioneer work [Ka] inspired further study of the geometry of theses spaces and of the analysis of the Laplacian thereon.