how can you tell the dimension of a manifold? one answer lies in studying the flow of heat on the manifold. {\em heat flow} is a smoothing process on riemannian manifolds, whose long-term behaviour is intimately linked to global geometry. however, the {\em short-time} smoothing behaviour is universal: it depends only upon the dimension of the manifold, and determines the dimension uniquely. \medskip in non-commutative geometry, the over-arching principal is to study a non-commutative algebra, pretend it is an algebra of smooth functions or differential operators on a {\em non-commutative manifold}, and import analytic and algebraic tools from global analysis to discover geometric facts about this manifold. \medskip while using heat flow is an excessively difficult way to determine the dimension of a manifold, it yields one approach to define dimension for non-commutative manifolds. in the context of {\em free probability} (one branch of non-commutative geometry concentrating on analytic properties of free groups), this leads, inexorably, to the somewhat comical-sounding conclusion that {\em all free groups have dimension $6$}. \medskip in this talk, i will outline those aspects of free probability which relate to heat kernel analysis, and make the connection between dimension and heat flow clear. i will also discuss recent joint work with roland speicher, showing that {\em all free semigroups have dimension $4$}.