Condensed-matter phenomena involving spontaneous spatial ordering can be realized using ultracold atomic condensates confined in multimode optical cavities. The atoms locally crystallize at antinodes of one of the cavity modes when the cavity is transversely pumped by a laser of sufficient intensity; the mode chosen for crystallization varies across the cavity, giving rise to dislocations and (in some regimes) geometrical frustration. The resulting ordered state is a “supersolid,” i.e., a superfluid that spontaneously breaks a continuous translational symmetry. The crystallization transition is a quantum phase transition, described by a theory involving a nested surface of low-energy excitations. We discuss the nature of fluctuations near the crystallization transition and low-energy excitations in the ordered state, and prospects for their experimental detection.