  This is an expanded version of our work [AN88], 1988, in Russian.
  We classify del Pezzo surfaces over C with log terminal singularities of
index \le 2. By classification, we understand a description of the intersection
graph of all exceptional curves on an appropriate (somewhat stronger than
minimal) resolution of singularities together with the subgraph of the curves
which are contracted to singular points.
  The final results are similar to classical results about classification of
non-singular del Pezzo surfaces and use the usual finite root systems. However,
the intermediate considerations use the theory of K3 surfaces (especially of K3
surfaces with non-symplectic involutions) and the theory of reflection groups
in hyperbolic spaces (especially or reflection groups of hyperbolic lattices).
  As an ``elementary'' application, our results permit one to classify sextics
in P^2 with simple singularities and a component of geometric genus \ge 2.
