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 consider a well-motivated class of theories based on the cubic Galileon action 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 comoving curvature modes throughout the NEC violating phase. We also discuss the relation between our procedure and earlier work.Introduction. Demonstrating that classical bounces are possible is an important milestone in constructing theories of the origin and evolution of the universe that avoid a big bang and its attendant singularity problem, or the invocation of large quantum gravity effects. The challenge has been to find examples that avoid instabilities or other pathologies so that it is possible to smoothly transit from a bounce to a homogeneous, isotropic and flat expanding universe that matches observations.The necessary conditions for a bounce can be understood by following the evolution of the Hubble parameter H ≡ȧ/a assuming Einstein gravity and a spatially flat Friedmann-Robertson-Walker (FRW) universe with metric ds 2 = −dt 2 + a 2 (t)dx i dx i (where dot denotes differentiation with respect to FRW time t): During a period of ordinary (not de Sitter or anti-de Sitter) contraction, H is becoming more negative as the scale factor a(t) is shrinking, and the total energy density ∝ H 2 is growing. During an ordinary expanding period, on the other hand, H is becoming less positive as a(t) is expanding, and the total energy density ∝ H 2 is shrinking. These two cosmological phases can only be connected classically if, towards the end of the ordinary contracting phase, H reverses its evolution and starts becoming less negative at a finite value of a, well before H 2 gets close to Planckian energies. During this 'bounce stage,' the increasing value of H eventually hits zero and continues to grow until it reaches a large positive value (well below the Planck scale but above the nucleosynthesis scale), at which point the bounce stage ends and H begins to decrease. In a flat FRW universe, a growing Hubble parameter (Ḣ > 0), as occurs during the bounce stage, corresponds to violating the null energy condition (NEC). Hence, we see the NEC violation is essential.To achieve NEC violation, various forms of stressenergy have been considered [1]. One of the best motivated examples is a scalar field described by a cubic Galileon action. An ordinary, canonical scalar field with quadratic kinetic term does not violate the NEC at all.