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The Markov point processes introduced by Ripley & Kelly are generalised by replacing fixed-range spatial interactions by interactions between neighbouring particles, where the neigh-bourhood relation may depend on the point configuration. The corresponding Hammersley-Clifford characterisation theorem is proved. The results are used to construct new models for point processes, multitype point processes and random processes of geometrical objects. Monte Carlo simulations of these models can be generated by running spatial birth-and-death processes.