This thesis discusses the random Euclidean bipartite matching problem i.e. the matching
problem between two different sets of points randomly generated on the Euclidean domain. The
presence of both randomness and Euclidean constraints makes the study of the average properties
of the solution highly relevant. The thesis reviews a number of known results about both
matching problems and Euclidean matching problems. It then goes on to provide a complete and
general solution for the one dimensional problem in the case of convex cost functionals and
moreover discusses a potential approach to the average optimal matching cost and its finite
size corrections in the quadratic case. The correlation functions of the optimal matching map
in the thermodynamical limit are also analyzed. Lastly using a functional approach the thesis
puts forward a general recipe for the computation of the correlation function of the optimal
matching in any dimension and in a generic domain.
