  We study a heavy piston of mass $M$ that moves in one dimension. The piston
separates two gas chambers, each of which contains finitely many ideal, unit
mass gas particles moving in $d$ dimensions, where $ d\geq 1$. Using averaging
techniques, we prove that the actual motions of the piston converge in
probability to the predicted averaged behavior on the time scale $M^ {1/2} $
when $M$ tends to infinity while the total energy of the system is bounded and
the number of gas particles is fixed. Neishtadt and Sinai previously pointed
out that an averaging theorem due to Anosov should extend to this situation.
  When $ d=1$, the gas particles move in just one dimension, and we prove that
the rate of convergence of the actual motions of the piston to its averaged
behavior is $\mathcal{O} (M^ {-1/2}) $ on the time scale $M^ {1/2} $. The
convergence is uniform over all initial conditions in a compact set. We also
investigate the piston system when the particle interactions have been
smoothed. The convergence to the averaged behavior again takes place uniformly,
both over initial conditions and over the amount of smoothing.
  In addition, we prove generalizations of our results to $N$ pistons
separating $N+1$ gas chambers. We also provide a general discussion of
averaging theory and the proofs of a number of previously known averaging
results. In particular, we include a new proof of Anosov's averaging theorem
for smooth systems that is primarily due to Dolgopyat.
