  This PhD thesis is organized as follows. In the first two chapters I will
review some basic notions of statistical physics of disordered systems, such as
random graph theory, the mean-field approximation, spin glasses and
combinatorial optimization. The replica method will also be introduced and
applied to the Sherrington-Kirkpatrick model, one of the simplest mean-field
models of spin-glasses. The second part of the thesis deals with mean-field
combinatorial optimization problems. The attention will be focused on the study
of finite-size corrections of random integer matching problems (chapter 3) and
fractional ones (chapter 4). In chapter 5 I will discuss a very general
relation connecting multi-overlaps and the moments of the cavity magnetization
distribution. In the third part we consider random Euclidean optimization
problems. I will start solving the traveling-salesman-problem (TSP) in one
dimension both in its bipartite and monopartite version (chapter 6). In chapter
7 I will discuss the possible optimal solutions of the 2-factor problem. In
chapter 8 I will solve the bipartite TSP in two dimensions, in the limit of
large number of points. Chapter 9 contains some conclusions.
