  We critically review several recent approaches to solving the two
cosmological constant problems. The "old" problem is the discrepancy between
the observed value of $\Lambda$ and the large values suggested by particle
physics models. The second problem is the "time coincidence" between the epoch
of galaxy formation $t_G$ and the epoch of $\Lambda$-domination $t_\L$. It is
conceivable that the "old" problem can be resolved by fundamental physics
alone, but we argue that in order to explain the "time coincidence" we must
account for anthropic selection effects. Our main focus here is on the
discrete-$\Lambda$ models in which $\Lambda$ can change through nucleation of
branes. We consider the cosmology of this type of models in the context of
inflation and discuss the observational constraints on the model parameters.
The issue of multiple brane nucleation raised by Feng {\it et. al.} is
discussed in some detail. We also review continuous-$\L$ models in which the
role of the cosmological constant is played by a slowly varying potential of a
scalar field. We find that both continuous and discrete models can in principle
solve both cosmological constant problems, although the required values of the
parameters do not appear very natural. M-theory-motivated brane models, in
which the brane tension is determined by the brane coupling to the four-form
field, do not seem to be viable, except perhaps in a very tight corner of the
parameter space. Finally, we point out that the time coincidence can also be
explained in models where $\Lambda$ is fixed, but the primordial density
contrast $Q=\delta\rho/\rho$ is treated as a random variable.

High Energy Physics - Theory;Astrophysics;General Relativity and Quantum Cosmology;High Energy Physics - Phenomenology