We study the nonequilibrium properties of directed Ising models with non-conserveddynamics, in which each spin is influenced by only a subset of its nearest neighbours. Wetreat the following models: (i) the one-dimensional chain; (ii) the two-dimensional squarelattice; (iii) the two-dimensional triangular lattice and (iv) the three-dimensionalcubic lattice. We raise and answer the question: (a) under what conditions is thestationary state described by the equilibrium Boltzmann–Gibbs distribution? Weshow that, for models (i), (ii) and (iii), in which each spin ‘sees’ only half of itsneighbours, there is a unique set of transition rates, namely with exponentialdependence in the local field, for which this is the case. For model (iv), we find thatany rates satisfying the constraints required for the stationary measure to beGibbsian should satisfy detailed balance, ruling out the possibility of directeddynamics. We finally show that directed models on lattices of coordination numberz≥8 with exponential rates cannot accommodate a Gibbsian stationary state. We conjecturethat this property extends to any form of the rates. We are thus led to the conclusion thatdirected models with Gibbsian stationary states only exist in dimensions one and two.We then raise the question: (b) do directed Ising models, augmented by Glauberdynamics, exhibit a phase transition to a ferromagnetic state? For the modelsconsidered above, the answers are open problems, with the exception of the simplecases (i) and (ii). For Cayley trees, where each spin sees only the spins further fromthe root, we show that there is a phase transition provided the branching ratio,q, satisfiesq≥3.