  We review our recent work on the universal (i.e. input state independent)
optimal quantum copying (cloning) of qubits. We present unitary transformations
which describe the optimal cloning of a qubit and we present the corresponding
quantum logical network. We also present network for an optimal quantum copying
``machine'' (transformation) which produces N+1 identical copies from the
original qubit. Here again the quality (fidelity) of the copies does not depend
on the state of the original and is only a function of the number of copies, N.
In addition, we present the machine which universaly and optimally clones
states of quantum objects in arbitrary-dimensional Hilbert spaces. In
particular, we discuss universal cloning of quantum registers.
