  In this paper, we study the 6d Little String Theory (LST) (the decoupled
theory on the worldvolume of $N$ NS5-branes) on curved manifolds, by using its
holographic duality to Type II string theory in asymptotically linear dilaton
backgrounds. We focus on backgrounds with a large number of Killing vectors
(namely, products of maximally symmetric spaces), without requiring
supersymmetry (we do not turn on any background fields except the metric). LST
is non-local so it is not obvious which spaces it can be defined on; we show
that holography implies that the theory cannot be put on negatively curved
spaces, but only on spaces with zero or positive curvature. For example, one
cannot put LST on a product of an anti-de Sitter space times another space,
without turning on extra background fields. On spaces with positive curvature,
such as $S^6$, $\mathbb{R}^2\times S^4$, $S^3\times S^3$, etc., we typically
find (for large $N$) dual holographic backgrounds which are weakly coupled and
weakly curved everywhere, so that they can be well-described by Type II
supergravity. In some cases more than one smooth solution exists for LST on the
same space, and they all contribute to the partition function. We also study
the thermodynamical properties of LST compactified on spheres, finding the
leading correction to the Hagedorn behavior of the spectrum, which is different
on curved space than on flat space. We discuss the holographic renormalization
procedure, which must be implemented in order to get a finite free energy for
the LST; we do not know how to implement it for general spaces, but we can (and
we do) implement it for the theory compactified on $S^4$.
