We will talk about the  construction and the profile of finite-time blow-up solutions to the semilinear heat equation on a compact Riemannian manifold. While most known singularity formation mechanisms rely on Euclidean concentration profiles, we develop a framework adapted to the geometric setting. In particular, we localizing the solution in local charts, and sum up information to gain control of the solution in all the manifold.  In addition, we prove a stability result under suitable perturbations of the initial data.