  We construct a logarithmic version of the Hilbert scheme, and more generally
the Quot scheme, of a simple normal crossings pair. The logarithmic Quot space
admits a natural tropicalisation called the space of tropical supports, which
is a functor on the category of cone complexes. The fibers of the map to the
space of tropical supports are algebraic. The space of tropical supports is
representable by ``piecewise linear spaces'', which are introduced here to
generalise fans and cone complexes to allow non--convex geometries. The space
of tropical supports can be seen as a polyhedral analogue of the Hilbert
scheme. The logarithmic Quot space parameterises quotient sheaves on
logarithmic modifications that satisfy a natural transversality condition. We
prove that the space it is a logarithmic algebraic space, is separated, and
universally closed. The logarithmic Hilbert space parameterizes families of
proper monomorphisms, and in this way is exactly analogous to the classical
Hilbert scheme. The new complexity of the space can then be viewed as stemming
from the complexity of proper monomorphisms in logarithmic geometry. Our
construction generalises the logarithmic Donaldson--Thomas space studied by
Maulik--Ranganathan to arbitrary rank and dimension, and the good degenerations
of Quot schemes of Li--Wu to simple normal crossings geometries.
