In this thesis, we consider two types of nonlinear Schrödinger equations (NLS), namely a class of nonlinear Schrödinger equations with mixed power nonlinearities in R^N and a class of Schrödinger-Poisson-Slater equations in R^3. These two types of NLS arise in various mathematical and physical models and have drawn wide attention in recent years.From the physical point of view, since, in addition to being a conserved quantity for the evolution equation, the mass often has a clear physical meaning; for instance, it represents the power supply in nonlinear optics, or the total number of atoms in Bose-Einstein condensation, etc, we focus on studying solutions having prescribed mass, namely normalized solutions. The existence, multiplicity, and stability issues of such solutions are considered in this thesis. We deal with both Sobolev sub-critical and Sobolev critical cases. Particular attention is paid to Sobolev critical cases in which many open problems remain. Since normalized solutions are found as critical points of an associated functional on a constraint, the main ingredients of our proofs are variational methods.The thesis consists of four chapters. Chapter 1 is an introduction to this thesis containing a brief presentation of issues treated and obtained results. In Chapter 2, we study Sobolev critical nonlinear Schrödinger equations with mixed power nonlinearities in R^N. In a situation where the associated functional is unbounded from below on the constraint, we prove the existence of two constrained critical points, one local minimizer, and one saddle point lying at a mountain pass level. We also show that the standing waves associated with the local minimizer are orbitally stable and the associated standing waves corresponding with saddle points lying at mountain pass levels are strongly unstable. The main difficulty is the presence of the Sobolev critical exponent. Concerning the local minimizer, it is not possible to use in a standard way the compactness by concentration approach developed by P. L. Lions. Even having the compactness, the global existence in evolution is still unknown. For the existence of the saddle point, we need a strict upper estimate of the associated mountain pass level that we derive using testing functions.In Chapter 3, we study Schrödinger-Poisson-Slater equations in R^3. We deal with some range of parameters under which the associated functional restricted on the constraint will sometimes be bounded, sometimes be unbounded. In the case where the geometric structure of the associated functional suggests the existence of local minima or global minima, we develop an argument to deal with both Sobolev sub-critical and Sobolev critical cases. In the case where the geometric structure of the associated functional suggests the existence of a saddle point, we need two different arguments to deal with Sobolev sub-critical and Sobolev critical cases. Finally, in Chapter 4, we present some concluding remarks about the two equations considered in this thesis and also we propose some open problems.