Mean curvature flow is the gradient flow of the volume functional of submanifolds which are smoothly immersed in a higher dimensional manifold. Along the flow, volume of the submanifold is decreasing. The flow satisfies a parabolic system of nonlinear partial differential equations. In this talk, we shall discuss some recent progress in mean curvature flow of submanifolds of codimension at least two (the non-hypersurface case). In particular, motivated by geometric and topological applications, we shall discuss the motion of real 2-dimensional symplectic surfaces in a Kahler-Einstein surface (complex 2-dimensional) and n-dimensional Lagrangian submanifolds in a Calabi-Yau manifold of complex dimension n.