For each $n \geq 1$, let $X_n = \{ X_n(t) : t \in [0,1] \}$ be an $L_2$ martingale in $D([0,1])$. Roughly speaking, the martingale central limit theorem states that if

\begin{itemize}

\item the jumps in the sample paths of $X_n$ become negligible (in a certain sense) as $n$ gets large and

\item the predictable (or optional) quadratic variations of the $X_n$ converge to a continuous, increasing and deterministic limit $h : [0,1] \rightarrow [0,{\infty})$ as $n$ goes to infinity

\end{itemize}

then the $X_n$ converge weakly in $D([0,1])$ to a Brownian motion $B_h$ which is ``stretched-out'' by $h$ (cf.\ Helland 82).

In this talk, we generalize the above result to the situtation in which the latter condition holds for a sequence of {\em{general quadratic variations}}---processes which resemble predictable/optional quadratic variation but are not necessarily adapted. An application to the convergence of set-indexed strong martingales will be briefly discussed.