Generally, the probability that a given atom in amaterial absorbs an X-ray photon in a single X-ray pulse is much less than unity for storage-ring-based X-ray sources, even for third-generation synchrotron radiation sources. This situation has changed dramatically with the arrival of X-ray free-electron lasers: In the micro-focus of an X-ray free-electron laser, saturation of X-ray photoabsorption is routinely achieved. The immediate consequence is that the overall behavior of matter under such extreme conditions is characterized by efficient multiphoton absorption via a sequence of single-photon absorption events combined with inner-shell decay cascades and collisional ionization processes. In this way, unusual, highly excited states of matter are formed. Focusing on free atoms, this article provides a theoretical framework for the description of X-ray-matter interactions. The nature of X-ray multiphoton physics is explained, and the theory is compared with experimental data on atoms.