Following work of A. Beilinson, S. Bloch, S. Lichtenbaum and others, Andrei Suslin and Vladimir Voevodsky have established much of the desired formalism expected of motivic cohomology for varieties over a field. The motivic chain complexes $\mathbf {Z}(n)$ of sheaves with transfers are constructed using sheaves of equidimensional cycles proper of relative dimension 0. Techniques of Suslin-Voevodsky permit the proofs of many good properties as well as some fundamental computations. Of particular importance is the close relationship between algebraic $K$-theory and motivic cohomology, the cohomology of $\mathbf {Z}(n)$.