"Power variation and time change" Ole E. Barndorff-Nielsen The Centre for Mathematical Physics and Stochastics (MaPhySto), University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark and Neil Shephard Nuffield College, Oxford OX1 1NF, UK. This paper provides limit distribution results for power variation, that is sums of powers of absolute increments, for certain types of time-changed Brownian motion and $\alpha $-stable processes. Special cases of these processes are stochastic volatility models used extensively in financial econometrics. Keywords: Power variation; r-variation; Realised variance; Semimartingales; Stochastic volatility; Time-change.