  In this paper, we study upper bounds for the topological complexity of the
total spaces of some classes of fibre bundles. We calculate a tight upper bound
for the topological complexity of an $n$-dimensional Klein bottle. We also
compute the exact value of the topological complexity of $3$-dimensional Klein
bottle. We describe the cohomology rings of several classes of generalized
projective product spaces with $\mathbb{Z}_2$-coefficients. Then we study the
LS-category and topological complexity of infinite families of generalized
projective product spaces. We reckon the exact value of these invariants in
many specific cases. We calculate the equivariant LS-category and equivariant
topological complexity of several product spaces equipped with
$\mathbb{Z}_2$-action.
