1 Introduction

2 Obstacle problems for plates
2.1 Unilateral displacements

3 The classical obstacle problem for the bi-Laplacian
3.1 Regularity of solutions
3.2 Regularity of the free boundary

4 The thin obstacle problem for the poly-Laplacian

5 The fractional Laplacian of order s<2

6 The extension problem for the higher order fractional Laplacian

7 The obstacle problem for a higher order fractional Laplacian

8 A two-phase boundary obstacle-type problem for the bi-Laplacian
8.1 Regularity of the minimizer
8.2 Regularity of the free boundary