Let $X$ be a process which is observed at the times $i\Delta_n$ for $i=0,1,\ldots,$. If $p>0$ the realized $p$-variation over the time interval $[0, t]$ is

V^n(p)_t=\sum_{i=1}^{[t/\Delta_n]}|X_{i\Delta_n}-X_{(i-1)\Delta_n}|^p.

The behavior of these $p$-variations when $\Delta_n ightarrow 0$ (and t is fixed) is now well understood, from the point of view of limits in probability (these are basically old results due to Lepingle) and also for the associated central limit theorem.
The aim of this talk is to review those results, as well as a few extensions (multipower variations, truncated variations). We will put some emphasis on the assumptions on $X$ which are needed, depending on the value of $p$.