In this paper we characterize flow-level stochastic stability for networks with non-convex or time-varying rate regions underresource allocation based on utility maximization. Similar to prior works on flow-level stability, we consider exogenous data arrivals with finite workloads. However, to model many realistic situations, the rate region, which constrains the feasibility of resource allocation, may be either non-convex or time-varying. When the rate region is fixed but non-convex, we derive sufficient and necessary conditions for stability, which coincide when the set of allocated rate vectors has continuous contours. When the rate region is time-varying according to some stationary, ergodic process, we derive the precise stability region. In both cases,the size of the stability region depends on the resource allocation policy, in particular, on the fairness parameter in -fair utility maximization. This is in sharp contrast with the substantial existing literature on stability under fixed and convex rate regions, in which the stability region coincides with the rate region for many utility-based resource allocation schemes, independently of the value of the fairness parameter. We further investigate the tradeoff between fairness and stability when rate region is non-convex or time-varying. Numerical examples of both wired and wireless networks are provided to illustrate the new stability regions and tradeoffs proved in the paper.