The first important contribution to the study of commutators is due to A. Wintner who in 1947 proved that the identity element 1 in a unital, normed algebra A is not a commutator, that is, there are no elements A and B such that I=AB - BA. In 1973 J.H. Anderson proved the remarkable result that there exists a bounded linear operator A such that I belongs to the closure in the norm topology of the range of the derivation AB-BA. The classical Brown-Pearcy characterization of the non commutators on $B(H)$ as the operators $\lambda I+K$, for $K\not= 0$ and $K$ a compact operator, is the natural motivation for Anderson's result. This leads us in this monograph to present a characterization of a commutator and to study the range of a derivation in relation to particular elements: Identity, commutant or the commutant of the adjoint of an operator. We study also the range and the kernel of an important class of elementary operators, in particular we study the range of a generalized derivation. This book is organized in a series of sections each of which develops some area of interest in operator theory, starting from a point accessible to advanced graduate students and researchers.