  In light of the recent observational data coming from the sky we have two
significant directions in the field of theoretical cosmology recently. First,
we are now able to make use of present observations, such as the Planck and
BICEP2 data, to examine theoretical predictions from the standard inflationary
$\Lambda$CDM which were made decades of years ago. Second, we can search for
new cosmological signatures as a way to explore physics beyond the standard
cosmic paradigm. In particular, a subset of early universe models admit a
nonsingular bouncing solution that attempts to address the issue of the big
bang singularity. These models have achieved a series of considerable
developments in recent years, in particular in their perturbative frameworks,
which made brand-new predictions of cosmological signatures that could be
visible in current and forthcoming observations. In this article we present two
representative paradigms of very early universe physics. The first is the
so-called new matter (or matter-ekpyrotic) bounce scenario in which the
universe starts with a matter-dominated contraction phase and transitions into
an ekpyrotic phase. In the setting of this paradigm, we propose some possible
mechanisms of generating a red tilt for primordial curvature perturbations and
confront its general predictions with recent cosmological observations. The
second is the matter-bounce inflation scenario which can be viewed as an
extension of inflationary cosmology with a matter contraction before inflation.
We present a class of possible model constructions and review its implications
on the current CMB experiments. Last, we review the significant achievements of
these paradigms beyond the inflationary $\Lambda$CDM model, which we expect to
shed new light on the direction of observational cosmology in the next decade
years.
