  We study admissible subcategories of derived categories of coherent sheaves
on del Pezzo surfaces and rational elliptic surfaces. Using a relation between
admissible subcategories and anticanonical divisors we prove the following
results. First, we classify all admissible subcategories of the projective
plane by showing that each is generated by a subcollection of a full
exceptional collection. Second, we show that the derived categories of del
Pezzo surfaces do not contain any phantom subcategories. This provides first
examples of varieties of dimension larger than one that have some nontrivial
admissible subcategories, but provably do not contain phantoms. We also prove
that any admissible subcategory supported set-theoretically on a smooth
(-1)-curve in a surface is generated by some twist of the structure sheaf of
that curve.
