  We present quasi-analytical and numerical calculations of Gaussian wave
packet solutions of the Schr\"odinger equation for two-dimensional infinite
well and quantum billiard problems with equilateral triangle, square, and
circular footprints. These cases correspond to N=3, N=4, and $N \to \infty$
regular polygonal billiards and infinite wells, respectively. In each case the
energy eigenvalues and wavefunctions are given in terms of familiar special
functions. For the first two systems, we obtain closed form expressions for the
expansion coefficients for localized Gaussian wavepackets in terms of the
eigenstates of the particular geometry. For the circular case, we discuss
numerical approaches. We use these results to discuss the short-time,
quasi-classical evolution in these geometries and the structure of wave packet
revivals. We also show how related half-well problems can be easily solved in
each of the three cases.
