  Integral invariants in maximally supersymmetric Yang-Mills theories are
discussed in spacetime dimensions $4\leq D\leq 10$ for $SU(k)$ gauge groups. It
is shown that, in addition to the action, there are three special invariants in
all dimensions. Two of these, the single- and double-trace $F^4$ invariants,
are of Chern-Simons type in $D=9,10$ and BPS type in $D\leq 8$, while the
third, the double-trace of two derivatives acting on $F^4$, can be expressed in
terms of a gauge-invariant super-$D$-form in all dimensions. We show that the
super-ten-forms for $D=10$ $F^4$ invariants have interesting cohomological
properties and we also discuss some features of other invariants, including the
single-trace $d^2 F^4$, which has a special form in $D=10$. The implications of
these results for ultra-violet divergences are discussed in the framework of
algebraic renormalisation.
