In this seminar I will introduce the notion of Wodzicki residue, also denoted by non-commutative residue, which was first introduced by Wodzicki in 1984 while studying the meromorphic continuation of the ζ-function for elliptic operators on compact manifold with boundary. The Wodzicki residue was independentely defined by Guillemin in 1985, in the equivalent version of Symplectic residue, in order to find a soft proof of the Weyl formula. It turns out to be the unique trace, up to a multiplication by a constant, on the algebra of classical pseudodifferential operators modulo smoothing operators, provided that the manifold has dimension d>1. In the last years, the interest in the study of Wodzicki residue increased due to its applications both in mathematics (non-commutative geometry) and mathematical physics (relations with Dixmier trace). I will discuss the concept of Wodzicki residue on compact manifold with boundary, for SG-calculus on R^d and for the SG-calculus on manifolds with cylindrical ends. Finally, as a joint work with Professor S. Coriasco, I will present an extension of the non-commutative residue on a certain class of non-compact manifolds called scattering manifolds.