  We study constraints imposed by two proposed string Swampland criteria on
cosmology. These criteria involve an upper bound on the range traversed by
scalar fields as well as a lower bound on $|\nabla_{\phi} V|/V$ when $V >0$. We
find that inflationary models are generically in tension with these two
criteria. Applying these same criteria to dark energy in the present epoch, we
find that specific quintessence models can satisfy these bounds and, at the
same time, satisfy current observational constraints. Assuming the two
Swampland criteria are valid, we argue that the universe will undergo a phase
transition within a few Hubble times. These criteria sharpen the motivation for
future measurements of the tensor-to-scalar ratio $r$ and the dark energy
equation of state $w$, and for tests of the equivalence principle for dark
matter.
