  We study oscillatory integrals in several variables with analytic, smooth, or
$C^k$ phases satisfying a nondegeneracy condition attributed to Varchenko. With
only real analytic methods, Varchenko's estimates are rediscovered and
generalized. The same methods are pushed further to obtain full asymptotic
expansions of such integrals with analytic and smooth phases, and finite
expansions with error assuming the phase is only $C^k$. The Newton polyhedron
appears naturally in the estimates; in particular, we show precisely how the
exponents appearing in the asymptotic expansions depend only on the geometry of
the Newton polyhedron of the phase. All estimates proven hold for oscillatory
parameter real and nonzero, not just asymptotically.
