Consider a semimartingale of the form $Y_t=Y_0+\int_0^ta_sds+\int_0^t \sigma_{s^-} dW_s$, where $a$ is a locally bounded predictable process and $\sigma$ (the "volatility") is an adapted right-continuous process with left limits and $W$ is a Brownian motion. We consider the realised bipower variation process $V(Y; r,s)_t^n=n^{\frac{r+s}{2}-1}\sum_{i=1}^{[nt]} | Y_{\frac{i}{n}}-Y_{\frac{i-1}{n}}| ^r | Y_{\frac{i+1}{n}}-Y_{\frac{i}{n}}| ^s$, where $r$ and $s$ are nonnegative reals with $r+s>0$. We prove that $V(Y;r,s)_t^n$ converges locally uniformly in time, in probability, to a limiting process $V(Y;r,s)_t$ (the "bipower variation process"). If further $\sigma$ is a possibly discontinuous semimartingale driven by a Brownian motion which may be correlated with $W$ and by a Poisson random measure, we prove a central limit theorem, in the sense that $\sqrt{n}(V(Y;r,s)^n-V(Y;r,s))$ converges in law to a process which is the stochastic integral with respect to some other Brownian motion $W'$, which is independent of the driving terms of $Y$ and $\sigma$. We also provide a multivariate version of these results, and a version in which the absolute powers are replaced by smooth enough functions.