  We propose a generic recipe for deforming extremal black holes into
non-extremal black holes and we use it to find and study the non-extremal
black-hole solutions of several N=2,d=4 supergravity models (SL(2,R)/U(1), CPn
and STU with four charges). In all the cases considered, the non-extremal
family of solutions smoothly interpolates between all the different extremal
limits, supersymmetric and not supersymmetric. This fact can be used to find
explicitly extremal non-supersymmetric solutions in the cases in which the
attractor mechanism does not completely fix the values of the scalars on the
event horizon and they still depend on the boundary conditions at spatial
infinity.
  We compare (supersymmetry) Bogomol'nyi bounds with extremality bounds, we
find the first-order flow equations for the non-extremal solutions and the
corresponding superpotential, which gives in the different extremal limits
different superpotentials for extremal black holes. We also compute the
"entropies" (areas) of the inner (Cauchy) and outer (event) horizons, finding
in all cases that their product gives the square of the moduli-independent
entropy of the extremal solution with the same electric and magnetic charges.
