  Common envelope evolution (CEE) is believed to be an important stage in the
evolution of binary/multiple stellar systems. Following this stage, the CE is
thought to be ejected, leaving behind a compact binary (or a merger product).
Although extensively studied, the CEE process is still little understood, and
although most binaries have non-negligible eccentricity, the effect of initial
eccentricity on the CEE has been little explored. Moreover, most studies assume
a complete circularization of the orbit by the CE onset, while observationally
such eccentricities are detected in many post-CE binaries. Here we use smoothed
particle hydro-dynamical simulations (SPH) to study the evolution of initially
eccentric ($0\le e\le0.95$) CE-systems. We find that initially eccentric
binaries only partially circularize. In addition, higher initial eccentricity
leads to a higher eccentricity following the end of the inspiral phase, with
eccentricities as high as 0.18 in the most eccentric cases, and even higher if
the initial peri-center of the orbit is located inside the star (e.g. following
a kick into an eccentric orbit, rather than a smooth transition). CEE of more
eccentric binaries leads to enhanced dynamical mass-loss of the CE compared
with more circular binaries, and depends on the initial closest approach of the
binary. We show that our results and the observed eccentricities of post-CE
binaries suggest that the typical assumptions of circular orbits following CEE
might potentially be revised. We expect post-CE eccentricities to affect the
delay time distributions of various transients such as supernovae, gamma-ray
bursts and gravitational-wave sources by up to tens of percents.
