  Lorenz attractors play an important role in the modern theory of dynamical
systems. The reason is that they are robust, i.e. preserve their chaotic
properties under various kinds of perturbations. This means that such
attractors can exist in applied models and be observed in experiments. It is
known that discrete Lorenz attractors can appear in local and global
bifurcations of multidimensional diffeomorphisms. However, to date, only
partial cases were investigated. In this paper bifurcations of homoclinic and
heteroclinic cycles with quadratic tangencies of invariant manifolds are
studied. A full list of such bifurcations, leading to the appearance of
discrete Lorenz attractors is provided. In addition, with help of numerical
techniques, it was proved that if one reverses time in the diffeomorphisms
described above, the resulting systems also have such attractors. This result
is an important step in the systematic studies of chaos and hyperchaos.
