  We study homological mirror symmetry for Del Pezzo surfaces and their mirror
Landau-Ginzburg models. In particular, we show that the derived category of
coherent sheaves on a Del Pezzo surface X_k obtained by blowing up CP^2 at k
points is equivalent to the derived category of vanishing cycles of a certain
elliptic fibration W_k:M_k\to\C with k+3 singular fibers, equipped with a
suitable symplectic form. Moreover, we also show that this mirror
correspondence between derived categories can be extended to noncommutative
deformations of X_k, and give an explicit correspondence between the
deformation parameters for X_k and the cohomology class [B+i\omega]\in
H^2(M_k,C).
