We review properties of two-dimensional matter-wave solitons, governed by the spinor system of Gross-Pitaevskii equations with cubic nonlinearity, including spin-orbit coupling and the Zeeman splitting. In contrast to the collapse instability typical for the free space, spin-orbit coupling gives rise to stable solitary vortices. These are semi-vortices with a vortex in one spin component and a fundamental soliton in the other, and mixed modes, with topological charges 0 and ±1 present in both components. The semivortices and mixed modes realize the ground state of the system, provided that the self-attraction in the spinor components is, respectively, stronger or weaker than their cross-attraction. The modes of both types degenerate into unstable Townes solitons when their norms attain the respective critical values, while there is no lower norm threshold for the stable modes existence. With the Galilean invariance lifted by the spin-orbit coupling, moving stable solitons can exist up to a mode-dependent critical velocity with two moving solitons merging into a single one as a result of collision. Augmenting the Rashba term by the Dresselhaus coupling has a destructive effect on these states. The Zeeman splitting tends to convert the mixed modes into the semivortices, which eventually suffer delocalization. Existence domains for the soliton families are reviewed in terms of experiment-related quantities.