  We study geometrical and dynamical properties of the so-called discrete
Lorenz-like attractors, that can be observed in three-dimensional
diffeomorphisms. We propose new phenomenological scenarios of their appearance
in one parameter families of such maps. We pay especially our attention to such
a scenario that can lead to period-2 Lorenz-like attractors. These attractors
have very interesting dynamical properties and we show that their crises can
lead, in turn, to the emergence of pseudohyperbolic discrete Lorenz shape
attractors of new types. We also show examples of all these attractors in
three-dimensional generalized H\'enon maps.
