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We theoretically investigate the Landau levels (LLs) and the magneto-transport and-optical properties in black phosphorus (BP) within linear-response theory by using a k·p theory and tight-binding method.For monolayer BP,we find that the LLs linearly depend both on the LL index n and magnetic field B at low field regime with different LL spacing in conduction and valence band,and strongly anisotropic wave-functions arising from the different anisotropic effective masses.An analytical expression for the LLs is obtained via solving the decoupled Hamiltonian,which agrees well with the numerical calculations.At high magnetic regime,a self-similar Hofstadter butterfly (HB) spectrum is obtained by using the TB model.The LLs of phosphorene nanoribbon depend strongly on the ribbon orientation due to the anisotropic hopping parameters.The Hall and the longitudinal conductance (resistance) clearly reveal the LL structure.For few layer BP,we obtain analytical expression for the LLs,which agrees well with the numerical calculations,and find that the LLs sub-linearly depend on the magnetic field and LL index.The analytical optical transition matrix elements reveal unusual selection rules for the inter-band (intra-band) optical transitions between the LLs △n =0,±2 (±1,± 3).The magneto-optical conductivity (MOC) shows strongly anisotropic behaviors of the band structure.The intra-band MOC can be used to determine the band parameters such as the effective masses and the inter-band coupling.Our results about the MOC can also be applied to the monolayer BP.The exchange enhancement of the g-factor in p-type BP quantum well (QW) is theoretically investigated by using the screened Hartree-Fock approximation.We demonstrate that the g factor of BP QW approaches to 2,i.e.,the g factor of bare electron in vacuum,due to huge built-in electric field,which leads to spatially separated electron and hole states.We also estimated the exchange-correlation interaction on the g-factor which shows maxima (minima) at odd (even) filling factors.The magnitude of exchange induced spin splitting,the amplitude and the shape of the g-factor oscillations are determined by both the screening of the electron-electron interaction and the LL broadening width.