  Despite its continued observational successes, there is a persistent (and
growing) interest in extending cosmology beyond the standard model,
$\Lambda$CDM. This is motivated by a range of apparently serious theoretical
issues, involving such questions as the cosmological constant problem, the
particle nature of dark matter, the validity of general relativity on large
scales, the existence of anomalies in the CMB and on small scales, and the
predictivity and testability of the inflationary paradigm. In this paper, we
summarize the current status of $\Lambda$CDM as a physical theory, and review
investigations into possible alternatives along a number of different lines,
with a particular focus on highlighting the most promising directions. While
the fundamental problems are proving reluctant to yield, the study of
alternative cosmologies has led to considerable progress, with much more to
come if hopes about forthcoming high-precision observations and new theoretical
ideas are fulfilled.
