  In this paper we investigate the nature of the transition from Abelian to
non-Abelian confinement (i.e. crossover vs. phase transition). To this end we
consider the basic N=2 model where non-Abelian flux tubes (strings) were first
found: supersymmetric QCD with the U(N) gauge group and N_f=N flavors of
fundamental matter (quarks). The Fayet-Iliopoulos term \xi triggers the squark
condensation and leads to the formation of non-Abelian strings. There are two
adjustable parameters in this model: \xi and the quark mass difference \Delta
m. We obtain the phase diagram on the (\xi, \Delta m) plane. At large \xi and
small \Delta m the world-sheet dynamics of the string orientational moduli is
described by N=2 two-dimensional CP(N-1) model. We show that as we reduce \xi
the theory exhibits a crossover to the Abelian (Seiberg-Witten) regime. Instead
of N^2 degrees of freedom of non-Abelian theory now only N degrees of freedom
survive in the low-energy spectrum. Dyons with certain quantum numbers condense
leading to the formation of the Abelian Z_N strings whose fluxes are fixed
inside the Cartan subalgebra of the gauge group. As we increase N this
crossover becomes exceedingly sharper becoming a genuine phase transition at N
=\infty.
