  We study the Z_N flux tubes and monopole confinement in deformed N=2* super
Yang-Mills theories. In order to do that we consider an N=4 super Yang-Mills
theory with an arbitrary gauge group G and add some N=2, N=1 and N=0
deformation terms. We analyze some possible vacuum solutions and phases of the
theory, depending on the deformation terms which are added. In the Coulomb
phase for the N=2* theory, G is broken to U(1)^r and the theory has monopole
solutions. Then, by adding some deformation terms, the theory passes to the
Higgs or color superconducting phase, in which G is broken to its center C_G.
In this phase we construct the Z_N flux tubes ansatz and obtain the BPS string
tension. We show that the monopole magnetic fluxes are linear integer
combinations of the string fluxes and therefore the monopoles can become
confined. Then, we obtain a bound for the threshold length of the
string-breaking. We also show the possible formation of a confining system with
3 different monopoles for the SU(3) gauge group. Finally we show that the BPS
string tensions of the theory satisfy the Casimir scaling law.
