  Duality transformations involving compactifications on timelike as well as
spacelike circles link M-theory, the 10+1-dimensional strong coupling limit of
IIA string theory, to other 11-dimensional theories in signatures 9+2 and 6+5
and to type II string theories in all 10-dimensional signatures. These theories
have BPS branes of various world-volume signatures, and here we construct the
world-volume theories for these branes, which in each case have 16
supersymmetries. For the generalised D-branes of the various type II string
theories, these are always supersymmetric Yang-Mills theories with 16
supersymmetries, and we show that these all arise from compactifications of the
supersymmetric Yang-Mills theories in 9+1 or 5+5 dimensions. We discuss the
geometry of the brane solutions and, for the cases in which the world-volume
theories are superconformally invariant, we propose holographically dual string
or M theories in constant curvature backgrounds. For product space solutions
$X\times Y$, there is in general a conformal field theory associated with the
boundary of $X$ and another with the boundary of $Y$.
