  We study the boundedness problem for maximal operators $\mathcal{M}$
associated to averages along families of hypersurfaces $S$ of finite type in
$\mathbb{R}^n.$ In this paper, we prove that if $S$ is a finite type
hypersurface which is of finite type $k$ at $x_0 \in \mathbb{R}^n$, then the
associated maximal operator is bounded on $L^p(\mathbb{R}^n)$ for $p>k.$ We
shall also consider a variable coefficient version of maximal theorem and we
obtain the same $L^p-$ boundedness result for $p>k.$
  We also discuss the consequence of this result. In particular, we verify a
conjecture by E. M. Stein and its generalization by A. Iosevich and E. Sawyer
on the connection between the decay rate of the Fourier transform of the
surface measure on $S$ and the $L^p-$ boundedness of the associated maximal
operator $\mathcal{M}.$
