Abstract One of the most extensively studied topics in Differential Geometry is minimal surfaces. This subject has connections with a wide range of mathematics areas, such as complex analysis, partial differential equations and topology. A classic problem in this context is the Plateau problem and recently much attention has been given to problems with the free boundary condition. This is a mini-course at master/beginning of PhD level. We are going to study problems about stability and classification for constant mean curvature (CMC) surfaces with the free boundary condition in certain spaces.