The authors consider backward stochastic differential equation (BSDE) with the generator \(g\) satisfying the standard conditions and define the families of nonlinear operators connected to the unique solution. These families are named quadratic g-evaluations and g-expectations. The main purpose of the paper is to prove some important properties of these families. The results obtained include the Doob-Meyer decomposition theorem, the optional sampling theorem, upcrossing inequality and Jensen’s inequality. It is also proved that the quadratic generator can be represented as the limit of the difference quotients of the corresponding g-evaluations, extending the similar results, obtained previously for linear growth case. With the help of these results, the reversed comparison theorem is also proved. The main tools are the techniques used in the study of quadratic BSDE and the properties of martingales with bounded mean oscillation.