We investigate the transport properties of topological insulator (TI) Bi0.83Sb0.17 nanowires. Single-crystal nanowire samples with diameters ranging from 75 nm to 1.1 μm are prepared using high frequency liquid phase casting in a glass capillary; cylindrical single crystals with (1011) orientation along the wire axis are produced. Bi0.83Sb0.17 is a narrow-gap semiconductor with an energy gap at the L point of the Brillouin zone, ΔE = 21 meV. The resistance of the samples increases with decreasing temperature, but a decrease in resistance is observed at low temperatures. This effect is a clear manifestation of TI properties (i.e., the presence of a highly conducting zone on the TI surface). We investigate the magnetoresistance of Bi0.83Sb0.17 nanowires at various magnetic field orientations. Shubnikov-de Haas oscillations are observed in Bi0.83Sb0.17 nanowires at T = 1.5 K, demonstrating the existence of high mobility (μS = 26700 - 47000 cm2V−1s−1) two-dimensional (2D) carriers in the surface areas of the nanowires, which are nearly perpendicular to the C3 axis. From the linear dependence of the nanowire conductance on nanowire diameter at T = 4.2 K, the square resistance Rsq of the surface states of the nanowires is obtained (Rsq = 70 Ohm). The oscillations of longitudinal magnetoresistance (MR) of 75 and 100-nm Bi0.83Sb0.17 nanowires with two periods ΔB1 and ΔB2 proportional to Φ0 and Φ0/2 were observed, where Φ0 = h/e is the flux quantum. A derivative of MR was measured at various inclined angles. In the range 0 – 60 degrees of inclined angle of magnetic field, the observed angle variation of the periods is in agreement with the theoretical dependence ΔB(α) = ΔB(0)/cosα of the flux quantization oscillations. However, the equidistant oscillations of MR exist in transverse magnetic fields under certain rotation angles. The observed effects are discussed.