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We study geometrical and dynamical properties of the so-called discrete Lorenz-like attractors. We show that such robustly chaotic (pseudohyperbolic) attractors can appear as a result of universal bifurcation scenarios, for which we give a phenomenological description and demonstrate certain examples of their implementation in one-parameter families of three-dimensional Hénon-like maps. We pay special attention to such scenarios 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 discrete Lorenz shape attractors of new types.

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