  We study the boundedness problem for maximal operators $\M$ associated to
smooth hypersurfaces $S$ in 3-dimensional Euclidean space. For $p>2,$ we prove
that if no affine tangent plane to $S$ passes through the origin and $S$ is
analytic, then the associated maximal operator is bounded on $L^p(\RR^3)$ if
and only if $p>h(S),$ where $h(S)$ denotes the so-called height of the surface
$S.$ For non-analytic finite type $S$ we obtain the same statement with the
exception of the exponent $p=h(S).$ Our notion of height $h(S)$ is closely
related to A. N. Varchenko's notion of height $h(\phi)$ for functions $\phi$
such that $S$ can be locally represented as the graph of $\phi$ after a
rotation of coordinates.
  Several consequences of this result are discussed. 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
$\M$, and a conjecture by Iosevich and Sawyer which relates the
$L^p$-boundedness of $\M$ to an integrability condition on $S$ for the distance
function to tangential hyperplanes, in dimension three.
  In particular, we also give ess. sharp uniform estimates for the Fourier
transform of the surface measure on $S,$ thus extending a result by V.N.
Karpushkin from the analytic to the smooth setting and implicitly verifying a
conjecture by V.I. Arnol'd in our context.
