  Let $M$ be a smooth compact surface, orientable or not, with boundary or
without it, $P$ either the real line $R^1$ or the circle $S^1$, and $Diff(M)$
the group of diffeomorphisms of $M$ acting on $C^{\infty}(M,P)$ by the rule
$h\cdot f\mapsto f \circ h^{-1}$, where $h\in Diff(M)$ and $f \in
C^{\infty}(M,P)$.
  Let $f:M \to P$ be a Morse function and $O(f)$ be the orbit of $f$ under this
action. We prove that $\pi_k O(f)=\pi_k M$ for $k\geq 3$, and $\pi_2 O(f)=0$
except for few cases. In particular, $O(f)$ is aspherical, provided so is $M$.
Moreover, $\pi_1 O(f)$ is an extension of a finitely generated free abelian
group with a (finite) subgroup of the group of automorphisms of the Reeb graph
of $f$.
  We also give a complete proof of the fact that the orbit $O(f)$ is tame
Frechet submanifold of $C^{\infty}(M,P)$ of finite codimension, and that the
projection $Diff(M) \to O(f)$ is a principal locally trivial $S(f)$-fibration.
