We consider a variety of semi-parametric models for a risky asset S = Log X and show how to robustly price and replicate a variety of path-dependent claims. The semi-parametric models we consider may exhibit both jumps and (possibly non-Markovian) stochastic volatility. Claims may depend on the terminal value of the log price X, its realized quadratic variation [X] and barrier-style events. This is joint work with Peter Carr and Roger Lee.