The exactly solvable Kitaev honeycomb model has inspired extensive research on novel states of matter induced by external magnetic fields, representing a major frontier in condensed matter physics, with candidate materials such as α-RuCl₃, Na₂Co₂TeO₆, and A₃Ni₂XO₆ (A = Li, Na; X = Sb, Bi). This review provides a systematic overview of the quantum and topological phases emerging in the Kitaev model and its extensions under magnetic fields. We first revisit the chiral spin liquid generated by a weak 111 magnetic field via third-order perturbation theory. The review then surveys exotic phases in low-dimensional Kitaev systems, including chiral solitons in one-dimensional chains characterized by staggered vector chirality, vortex gases and vortex glasses in quasi-one-dimensional ladders exhibiting random flux distributions, and a constrained classical spin liquid phase protected by local constraints. For the spin-1/2 antiferromagnetic Kitaev model, we compare predictions from various quantum many-body methods, which reveal divergent interpretations of the intermediate field-induced phase—ranging from a gapless U(1) spin liquid, a Majorana metal with logarithmically divergent density of states, to a gapped Abelian chiral spin liquid. We further discuss the spin-1 Kitaev model, where the zero-field ground state is consistently identified as a gapped \mathbbZ_2 spin liquid by tensor-network methods, and its magnetic-field-induced phase transitions. Extended Kitaev models exhibit even richer phenomena, such as Dirac and chiral spin liquids, a nematic paramagnet breaking C₃ lattice rotational symmetry, and field-induced spin-flop transitions with emergent U(1) symmetry. In the high-field polarized phase, topological magnon excitations emerge, giving rise to a finite thermal Hall conductivity that exhibits a strong dependence on the in-plane field direction, with sign changes and vanishing under certain symmetries. Moreover, the star lattice hosts high-Chern-number magnon bands and topological flat bands; notably, the thermal Hall conductivity exhibits nodal behavior with sign reversal at phase-transition points and approaches zero without sign change at flat-band points. This review provides a theoretical reference for understanding field-induced phenomena in Kitaev candidate materials and offers guidance for future experimental and numerical investigations.