The recent robust and homogeneous analysis of the world's supernova distance-redshift data, together with cosmic microwave background and baryon acoustic oscillation data-provides a powerful tool for constraining cosmological models. Here we examine particular classes of scalar field, modified gravity, and phenomenological models to assess whether they are consistent with observations even when their behavior deviates from the cosmological constant ?. Some models have tension with the data, while others survive only by approaching the cosmological constant, and a couple are statistically favored over ? cold dark matter. Dark energy described by two equation-of-state parameters has considerable phase space to avoid ? and next-generation data will be required to constrain such physics, with the level of complementarity between probes varying with cosmology.