A surface in R^3 is minimal if its area does not change (to first order) under a small deformation. The study of such surfaces dates back to the 18th century (first studied by Euler and Lagrange). An important theme in the study of minimal surfaces is the classification of minimal surfaces under certain conditions. For example, the famous Bernstein problem asks which minimal surfaces can be written as graphs over a plane. Closely related to the Bernstein problem is the study of minimal surfaces satisfying stronger conditions that criticality of the area functional, such as stability (a surface that the second derivative test), area minimality (a surface that is the global minimum of area) or bounded Morse index. I will describe some recent results in this direction (including some results obtained with Chao Li and Davi Maximo).