  One of the fundamental questions of theoretical cosmology is whether the
universe can undergo a non-singular bounce, i.e., smoothly transit from a
period of contraction to a period of expansion through violation of the null
energy condition (NEC) at energies well below the Planck scale and at finite
values of the scale factor such that the entire evolution remains classical. A
common claim has been that a non-singular bounce either leads to ghost or
gradient instabilities or a cosmological singularity. In this letter, we
examine cubic Galileon theories and present a procedure for explicitly
constructing examples of a non-singular cosmological bounce without
encountering any pathologies and maintaining a sub-luminal sound speed for
co-moving curvature modes throughout the NEC violating phase. We also discuss
the relation between our procedure and earlier work.
