New asymptotic methods for the analysis of queueing systems are introduced and applied to state-dependent M/G/1 queues. The methods are used to compute approximations to the stationary density of the queue length, the mean length of a busy period, the mean number of customers served during a busy period as well as other quantities of interest. We obtain results that are superior to those obtained from diffusion approximations in that they are uniformly valid for all values of the traffic intensity while diffusion approximations are adequate only when this quantity is close to one. When specialized to state-independent queues, our approximations are shown to agree with the asymptotic expansions of known exact results. Finally, we show that the behavior of the state-dependent systems is markedly different from that of the corresponding state-independent systems.