  This essay is a brief review of the recent studies of non-singular
cosmological scenarios with bounce and Genesis and their stability in a
subclass of scalar-tensor theories with higher derivatives -- beyond Horndeski
theories. We discuss the general results of stability analysis of the
non-singular cosmological solutions in beyond Horndeski theories, as well as
other closely related topics: 1) the no-go theorem, which is valid in the
general Horndeski theories but not in their extensions, 2) singularities in
disformal transformations relating beyond Horndeski theories with general ones,
3) healthy behaviour of the scalar sector in the unitary gauge despite
divergencies of coefficients in the quadratic action for perturbations
("$\gamma$-crossing"). We describe several specific examples of bouncing
cosmologies and models with Genesis epoch which have neither ghosts nor
gradient instabilities among the linearized perturbations about the homogeneous
isotropic background during entire evolution.
