Let $M$ be a smooth finite dimensional manifold, how restrictive is it to require $M$ to admit a smooth action by a finite dimensional Lie group $G$?

Related questions/approaches: Of course we need $\mathrm{dim}(G) \geq \mathrm{dim}(M)$, are there any results relating the minimal dimension of a Lie group acting transitively to that of $M$, perhaps in special cases?

Going in the other direction, any criteria which easily allow to say that a given smooth manifold does not admit a transitive group action?

EDIT: in the first question I meant to write *transitive* smooth group action.