The fundamental capacity limits of intelligent reflecting surface (IRS)-assisted multi-user wireless communication systems are investigated in this paper. Specifically, the capacity and rate regions for both capacity-achieving non-orthogonal multiple access (NOMA) and orthogonal multiple access (OMA) transmission schemes are characterized under the constraints of a maximum number of IRS reconfiguration times. For NOMA, the ideal case with asymptotically large number of IRS reconfiguration times is considered, where the optimal solution is obtained by employing the Lagrange duality method. Inspired by this result, an inner bound of the capacity region for the general case with a finite number of IRS reconfiguration times is derived. For OMA, the optimal transmission strategy for the ideal case is to serve each individual user alternatingly with its effective channel power gain maximized. Based on this result, a rate region inner bound for the general case is derived. Finally, numerical results are provided to show that: i) a significant capacity and rate region improvement can be achieved by using IRS; ii) the capacity gain can be further improved by dynamically configuring the IRS reflection matrix.