Operator-limit distributions are those when partial sums are normalized by arbitrary linear operators (matrices in Euclidean spaces) instead of numbers (i.e. multiplies of identity operator). In a such way we obtain operator-stable and operator-selfdecomposable measures. Both classes depend on an operator, say Q, which in the case of operator-stable measures plays the role of an exponent. This allows to select the norming operators from a one-parameter group \(t^ Q\), \(t>0\). Finally, by induction we get Urbanik classes of limit distributions \(L_ m(Q)\) such that \[ ID\supseteq L_ 0(Q)\supseteq L_ 1(Q)\subseteq L_ 2(Q)\supseteq...\supseteq L_{\infty}(Q)\supseteq S(Q), \] where ID is the class of all infinitely divisible measures, \(L_ 0(Q)\) consists of operator-selfdecomposable ones and S(Q) is the class of Q-stable measures. \(L_{\infty}(Q)\) is defined as the intersection of all \(L_ m(Q)\), \(m\geq 1\) and its elements are called completly operator- selfdecomposable measures. The authors study elements from \(L_{\infty}(Q)\) in finite dimensional spaces. They prove, for instance, that \(L_{\infty}(Q)\) can be viewed as the smallest closed convolution semigroup containing the class S(Q), cf. theorem 7.3.