  We investigate nonequilibrium steady states in a class of one-dimensional
diffusive systems that can attain negative absolute temperatures. The cases of
a paramagnetic spin system, a Hamiltonian rotator chain and a one-dimensional
discrete linear Schr\"odinger equation are considered. Suitable models of
reservoirs are implemented to impose given, possibly negative, temperatures at
the chain ends. We show that a phenomenological description in terms of a
Fourier law can consistently describe unusual transport regimes where the
temperature profiles are entirely or partially in the negative-temperature
region. Negative-temperature Fourier transport is observed both for
deterministic and stochastic dynamics and it can be generalized to coupled
transport when two or more thermodynamic currents flow through the system.
