Let \(G\) be a group, \(S=M_n(D)\) the \(n\times n\) matrix ring over a division ring \(D\), and \(E_{ij}\), \(1\leq i,j\leq n\), the matrix units. A grading \(S=\bigoplus_{g\in G}S_g\) on the ring \(S\) is called elementary if there exists a set \((g_1,\dots,g_n)\in G^n\) such that \(E_{ij}\in S_g\Leftrightarrow g=g_i^{-1}g_j\), and \(D\subseteq S_e\). A result of M. V. Zaicev and S. K. Sehgal [Mosc. Univ. Math. Bull. 56, No. 3, 21-24 (2001); translation from Vestn. Mosk. Univ., Ser. I 2001, No. 3, 21-25 (2001; Zbl 1054.16032)] asserts that any simple Artinian ring \(R=\bigoplus_{g\in G}R_g\) with a finite \(G\)-grading by a torsion-free group \(G\) is isomorphic as graded ring to a matrix ring \(M_n(D)\) with an elementary \(G\)-grading. In the present paper, using this result the authors give a complete description of all types of finite gradings on a semisimple Artinian ring if the grading group is torsion-free. If \(R\) is a matrix algebra over an algebraically closed field \(F\) and \(G\) is an Abelian group they describe all \(G\)-gradings on \(R\). In the case of an Abelian group \(G\) they also classify all finite dimensional graded simple algebras and finite dimensional graded division algebras over an algebraically closed field of characteristic zero.