Random Obstacle Problems : École dÉté de Probabilités de Saint Flour XLV 2015 Studying the fine properties of solutions to Stochastic (Partial) Differential Equations with reflection at a boundary this book begins with a discussion of classical one dimensional diffusions as the reflecting Brownian motion devoting a chapter to Bessel processes and moves on to function valued solutions to SPDEs. Inspired by the classical stochastic calculus for diffusions which is unfortunately still unavailable in infinite dimensions it uses integration by parts formulae on convex sets of paths in order to describe the behaviour of the solutions at the boundary and the contact set between the solution and the obstacle. The text may serve as an introduction to space time white noise SPDEs and monotone gradient systems. Numerous open research problems in both classical and new topics are proposed 1 Introduction 2 The reflecting Brownian motion 3 Bessel processes 4 The stochastic heat equation 5 Obstacle problems 6 Integration by Parts Formulae 7 The contact set References