  We investigate convergence of martingales adapted to a given filtration of
finite $\sigma$-algebras. To any such filtration we associate a canonical
metrizable compact space $K$ such that martingales adapted to the filtration
can be canonically represented on $K$. We further show that (except for trivial
cases) typical martingale diverges at a comeager subset of $K$. `Typical
martingale' means a martingale from a comeager set in any of the standard
spaces of martingales. In particular we show that a typical $L^1$-bounded
martingale of norm at most one converges almost surely to zero and has maximal
possible oscillation on a comeager set.
