Geometric flows such as Ricci flow and mean curvature flow are nonlinear parabolic PDEs that tend to develop singularities in finite time. A useful approach to analyzing singularities is the technique of matched asymptotics, which can provide detailed and precise information including the rate of curvature blow-up, the set of points where singularity forms, and the behavior of the solution in a space-time neighborhood of that singularity. In this talk, we will survey the results concerning the asymptotic shapes of neckpinch singularities in Ricci flow and mean curvature flow.