  Using U-duality, the properties of the matrix theories corresponding to the
compactification of M-theory on $T^d$ are investigated. The couplings of the
$d+1$ dimensional effective Super-Yang-Mills theory to all the M-theory moduli
is deduced and the spectrum of BPS branes in the SYM gives the corresponding
spectrum of the matrix theory.Known results are recovered for $d\le 5$ and
predictions for $d>5$ are proposed. For $d>3$, the spectrum includes $d-4$
branes arising from YM instantons, and U-duality interchanges momentum modes
with brane wrapping modes.For $d=6$, there is a generalised $\th $-angle which
couples to instantonic 3-branes and which combines with the SYM coupling
constant to take values in $SL(2,\R)/U(1)$, acted on by an $SL(2,\Z)$ subgroup
of the U-duality group $E_6(\Z)$. For $d=4,7,8$, there is an $SL(d+1)$
symmetry, suggesting that the matrix theory could be a scale-invariant $d+2$
dimensional theory on $T^{d+1} \times \R$ in these cases, as is already known
to be the case for $d=4$; evidence is found suggesting this happens for $d=8$
but not $d=7$.
