  We present improved approximation algorithms in stochastic optimization. We
prove that the multi-stage stochastic versions of covering integer programs
(such as set cover and vertex cover) admit essentially the same approximation
algorithms as their standard (non-stochastic) counterparts; this improves upon
work of Swamy \& Shmoys which shows an approximability that depends
multiplicatively on the number of stages. We also present approximation
algorithms for facility location and some of its variants in the $2$-stage
recourse model, improving on previous approximation guarantees. We give a
$2.2975$-approximation algorithm in the standard polynomial-scenario model and
an algorithm with an expected per-scenario $2.4957$-approximation guarantee,
which is applicable to the more general black-box distribution model.
