Open quantum walks (OQW) are formulated as quantum Markov chains on graphs. It is shown that OQWs contain as a limit classical random walks. A suggested physical realization procedure for OQW can recover unitary quantum walks. A straightforward unravelling of the OQW allows for stochastic simulations in terms of quantum trajectories. It is shown that OQWs are a very useful tool for the formulation of dissipative quantum computing algorithms and for dissipative quantum state preparation. In particular, single qubit gates and the CNOT-gate are implemented as OQWs on fully connected graphs. Also, dissipative quantum state preparation of arbitrary single qubit states and of all two-qubit Bell-states is demonstrated. Finally, the discrete time version of dissipative quantum computing is shown to be more efficient if formulated in the languageof OQWs.