  We prove lower bounds for the error of optimal cubature formulae for
$d$-variate functions from Besov spaces of mixed smoothness
$B^{\alpha}_{p,\theta}({\mathbb G}^d)$ in the case $0 < p, \theta \le \infty$
and $\alpha > 1/p$, where ${\mathbb G}^d$ is either the $d$-dimensional torus
${\mathbb T}^d$ or the $d$-dimensional unit cube ${\mathbb I}^d$. We prove
upper bounds for QMC methods of integration on the Fibonacci lattice for
bivariate periodic functions from $B^{\alpha}_{p,\theta}({\mathbb T}^2)$ in the
case $1\leq p \leq \infty$, $0 < \theta \leq \infty$, $\alpha>1/p$. A
non-periodic modification of this classical formula yields upper bounds for
$B^{\alpha}_{p,\theta}({\mathbb I}^2)$ if $1/p<\alpha<1+1/p$. In combination
these results yield the correct asymptotic error of optimal cubature formulae
for functions from $B^{\alpha}_{p,\theta}({\mathbb G}^2)$ and indicate that a
corresponding result is most likely also true in case $d>2$. This is compared
to the correct asymptotic of optimal cubature formulae on Smolyak grids which
results in the observation that any cubature formula on Smolyak grids is never
optimal for the general setting.
