In multiaccess wireless systems, dynamic allocation of resources such as transmit power, bandwidths, and rates is an important means to deal with the time-varying nature of the environment. In this two-part paper, we consider the problem of optimal resource allocation from an information-theoretic point of view. We focus on the multiaccess fading channel with Gaussian noise, and define two notions of capacity depending on whether the traffic is delay-sensitive or not. In part I, we characterize the throughput capacity region which contains the long-term achievable rates through the time-varying channel. We show that each point on the boundary of the region can be achieved by successive decoding. Moreover, the optimal rate and power allocations in each fading state can be explicitly obtained in a greedy manner. The solution can be viewed as the generalization of the water-filling construction for single-user channels to multiaccess channels with arbitrary number of users, and exploits the underlying polymatroid structure of the capacity region. In part II, we characterize a delay-limited capacity region and obtain analogous results.