  We make a number of conjectures about the geometry of continuous moduli
parameterizing the string landscape. In particular we conjecture that such
moduli are always given by expectation value of scalar fields and that moduli
spaces with finite non-zero diameter belong to the swampland. We also
conjecture that points at infinity in a moduli space correspond to points where
an infinite tower of massless states appear, and that near these regions the
moduli space is negatively curved. We also propose that there is no non-trivial
1-cycle of minimum length in the moduli space. This leads in particular to the
prediction of the existence of a radially massive partner to the axion. These
conjectures put strong constraints on inflaton potentials that can appear in a
consistent quantum theory of gravity. Our conjectures are supported by a number
of highly non-trivial examples from string theory. Moreover it is shown that
these conditions can be violated if gravity is decoupled.
