It is an open question to determine which Hamiltonian isotopy classes of Lagrangians in a Calabi-Yau manifold have a special Lagrangian representative. One approach is to follow the steepest descent of area, i.e. the mean curvature flow, which preserves the Lagrangian condition. But in general such a flow will develop singularities in finite time, and it has been open how to continue the flow past singularities. We will give an introduction to the problem and explain recent advances where we show that in the simplest possible situation, i.e. the Lagrangian mean curvature flow of surfaces, when the singularity is the special Lagrangian union of two transverse planes, then the flow forms a “neck pinch”, and can be continued past the singularity. This is joint work with Jason Lotay and Gábor Székelyhidi.