We extend the notion of static coherent risk measures generated by distortion functions to the dynamic discrete time setup. We establish some fundamental properties of such risk measures, and consequently, study the dynamic coherent acceptability indices generated by families of dynamic coherent risk measures generated by distortion functions. A major part of this work is dedicated to the time consistency of these classes of measures. We show that for a reasonably large class of distortions, the corresponding coherent risk measures are not strongly time consistent (recursive) and satisfy only a weaker form of time consistency. Finally, we characterize the risk and performance measures generated by distortions in terms of dynamic weighted value at risk measures. This is joint work with T. R. Bielecki and H. Liu