  Milnor manifolds are a class of certain codimension-$1$ submanifolds of the
product of projective spaces. In this paper, we study the LS-category and
topological complexity of these manifolds. We determine the exact value of the
LS-category and in many cases, the topological complexity of these manifolds.
We also obtain tight bounds on the topological complexity of these manifolds.
It is known that Milnor manifolds admit $\mathbb{Z}_2$ and circle actions. We
compute bounds on the equivariant LS-category and equivariant topological
complexity of these manifolds. Finally, we describe the mod-$2$ cohomology
rings of some generalized projective product spaces corresponding to Milnor
manifolds and use this information to compute the bound on LS-category and
topological complexity of these spaces.
