Nanoscale superlattices with uniaxial ferromagnetic layers antiferromagnetically coupled through non-magnetic spacers are recently used as components of magnetoresistive and recording devices. In the last years intensive experimental investigations of these artificial antiferromagnets have revealed a large variety of surface induced reorientational effects and other remarkable phenomena unknown in other magnetic materials. In this paper we review and generalize theoretical results, which enable a consistent description of the complex magnetization processes in antiferromagnetic multilayers, and we explain the responsible physical mechanism. The general structure of phase diagrams for magnetic states in these systems is discussed. In particular, our results resolve the long standing problem of a "surface spin-flop" in antiferromagnetic layers. This explains the different appearance of field-driven reorientation transitions in systems like Fe/Cr (001) and (211) [2,3,4,5,6] and references in [7,8]. These synthetic antiferromagnets are of great interest in modern nanomagnetism, in particular due to their application in spin electronics [9] and high-density recording technologies [10].In view of their magnetic states and field-induced reorientation transitions, these antiferromagnetically coupled superlattices can be separated into two groups: (1) Systems with magnetization in the film plane and low (higherorder) anisotropies only, e.g., multilayers grown on (001) faces of cubic substrates [3] with a four-fold anisotropy owing to the magneto-crystalline anisotropies of the materials (for further references and a survey of their magnetic properties, see [8]). In these high symmetry systems, magnetic states are mainly determined under competing influence of bilinear and biquadratic exchange interactions and the intrinsic magnetic anisotropy. For this type of multilayers with a fully compensated antiferromagnetic collinear ground state, the magnetization processes generally have a simple character, in particular in the low anisotropy limit no reorientation transition occurs with fields in direction of easy axes [8]. (2) The other group of the synthetic antiferromagnets own an often sizeable uniaxial anisotropy. This group includes superlattices with intrinsic or induced in-plane uniaxial anisotropy, e.g. multilayers on (110) and (211) faces of cubic substrates [2,4], or nanostructures with perpendicular anisotropy [5]. Here, an interplay between the uniaxial anisotropy and the confining geometry of the multilayers determines their magnetic properties and gives rise to effects such as surface spin-flops [2, 4], field induced cascades of magnetization jumps [5,11] and multidomain structures [6,8].Theoretical investigations of such systems have a long history [12] and a large number of results on the magnetic states (mostly within simplified models and by numerical methods) have been obtained [12,13,14]. However, these findings were often restricted to specific values of magnetic parameters and led to conflicting conclusio...