This book chapter is published open access.
On a Riemannian manifold, , the heat kernel is a smooth function on , , and the shape of this function depends on the properties of . This article pays particular attention to the long-time, large-scale behavior of the heat kernel and its relation to the global geometry of . When does the heat kernel look like a bell curve? If it does not, what does it look like and why? To answer such questions, one needs tools to obtain sharp two-sided estimates for the heat kernel in terms of the time variable and basic geometric quantities depending on . Under what assumptions on , can one hope to obtain such bounds?