Given a smooth map \(v_0 \colon M \rightarrow N\) between two smooth manifolds, is it possible to find a harmonic map (i.e., a stationary point for the Dirichlet energy) \(v \colon M \rightarrow N\) which is homotopic to \(v_0\)? This is the question that interested Eells and Sampson back in the sixties. To solve this problem positively, they introduced the Harmonic Map Heat Flow (HMHF) equation and extracted the harmonic map along the heat flow. If \(M\) is a surface, it turns out that the existence of such harmonic maps is closely related to blow-ups of the HMHF: singularities form by bubbling off harmonic spheres along the flow. In this talk I will go over the original idea of Eells and Sampson, as well as classical and recent results concerning the blow-up description of the HMHF. I will also present some ingredients which I recently used to exclude multi-bubbles in the case of finite-time blow-up for the HMHF from \(B^2\) to \(S^2\).