In this chapter we will describe the main methods of analysis used in this book. In Section 1 we will introduce function spaces and operators; we will discuss Fredholm property and solvability conditions for linear operators and properness of nonlinear operators. Topological degree will be discussed in Section 2. Here, as in the previous section, we use the results of Volume 1. In Section 3 we will recall positiveness and comparison theorems for scalar second-order equations and monotone systems of equations. They will be often used to get estimates of solutions and some their properties. Finally, Section 4 deals with some spectral properties. We will formulate the theorems which establish the relation between stability of solutions and location of the spectrum of linearized operators.