  The much-discussed swampland conjectures suggest significant constraints on
the properties of string theory landscape and on the nature of the multiverse
that this landscape can support. The conjectures are especially constraining
for models of inflation; in particular, they exclude the existence of de Sitter
(dS) vacua. If the conjectures are false and dS vacua do exist, it still
appears that their construction in string theory requires a fair amount of
fine-tuning, so they may be vastly outnumbered by AdS vacua. Here we explore
the multiverse structure suggested by these considerations. We consider two
scenarios: (i) a landscape where dS vacua are rare and (ii) a landscape where
dS vacua do not exist and the dS potential maxima and saddle points are not
flat enough to allow for the usual hilltop inflation, even though slow-roll
inflation is possible on the slopes of the potential. We argue that in both
scenarios inflation is eternal and all parts of the landscape that can support
inflation get represented in the multiverse. The spacetime structure of the
multiverse in such models is nontrivial and is rather different from the
standard picture.
