  We study spatially flat bouncing cosmologies and models with the early-time
Genesis epoch in a popular class of generalized Galileon theories. We ask
whether there exist solutions of these types which are free of gradient and
ghost instabilities. We find that irrespectively of the forms of the Lagrangian
functions, the bouncing models either are plagued with these instabilities or
have singularities. The same result holds for the original Genesis model and
its variants in which the scale factor tends to a constant as $t\to -\infty$.
The result remains valid in theories with additional matter that obeys the Null
Energy Condition and interacts with the Galileon only gravitationally. We
propose a modified Genesis model which evades our no-go argument and give an
explicit example of healthy cosmology that connects the modified Genesis epoch
with kination (the epoch still driven by the Galileon field, which is a
conventional massless scalar field at that stage).
