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In this talk, we will introduce the special Lagrangian and Lagrangian mean curvature type equations. We will discuss some recent developments on apriori estimates, regularity, and the existence of solutions of Lagrangian mean curvature type equations and some applications. We will also briefly mention some recent results on the regularity of solutions of certain fourth-order PDEs (such as the Hamiltonian stationary equation), which are critical points of variational integrals of the Hessian of a scalar function. Examples include volume functionals on Lagrangian submanifolds. This is based on joint works with Connor Mooney and Ravi Shankar.