Ultracold molecules give rise to new types of correlated matter driven by their strong dipolar interactions and numerous rotational states. I will describe how one can exploit these rotational states as a "synthetic dimension", an effective spatial dimension in addition to the real physical ones. Hundreds of fully tunable, synthetic lattice sites are feasible, in contrast to atoms, where synthetic dimensions are typically restricted to three sites. We have found that molecules with synthetic dimensions show phenomena that do not occur elsewhere in nature. One example occurs by freezing molecules in place in a deep lattice and applying microwaves to couple the rotational states. Several phases of matter arise, including one in which the system spontaneously reduces its dimension, forming a fluctuating quantum string or membrane on which a strongly interacting condensate lives. I will describe this mechanism and the properties of the quantum membrane.