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The capacity region of the finite-state multiple-access channel (FS-MAC) with feedback that may be an arbitrary time-invariant function of the channel output samples is considered. We characterize both an inner and an outer bound for this region, using Massey's directed information. These bounds are shown to coincide, and hence yield the capacity region, of indecomposable FS-MACs without feedback and of stationary and indecomposable FS-MACs with feedback, where the state process is not affected by the inputs. Though multiletter in general, our results yield explicit conclusions when applied to specific scenarios of interest. For example, our results allow us to do the following. 1. Identify a large class of FS-MACs, that includes the additive mod2 noise MAC where the noise may have memory, for which feedback does not enlarge the capacity region. 2. Deduce that, for a general FS-MAC with states that are not affected by the input, if the capacity (region) without feedback is zero, then so is the capacity (region) with feedback. 3. Deduce that the capacity region of a MAC that can be decomposed into a multiplexer concatenated by a point-to-point channel (with, without, or with partial feedback), the capacity region is given by Sigmam Rm les C, where C is the capacity of the point to point channel and m indexes the encoders. Moreover, we show that for this family of channels source-channel coding separation holds.