We will discuss Gradient flow structures in parabolic type PDEs and interface motions. Such structure introduces physical interpretations to time-dependent PDEs in terms of energy dissipation, and allows a weak notion of global-time solutions to nonlinear PDE problems. In broader terms, we will discuss minimizing movements, which approximates dissipative PDE systems with a discrete-time optimization problem. Lecture 1 will be a general introduction. Lecture 2 and 3 will discuss specific examples of minimizing movements: the mean curvature flow and nonlinear diffusions with Darcy's law.