Mesoscopic systems have been investigated for many decades. The properties of one dimensional system or quantum wire have been investigated theoretically; the case of Bi nanowires was studied by Hicks and Dresselhaus [1]. Bi has a Fermi surface consisting of small hole and electron pockets, therefore, the Fermi wavelength λF is very long (about 50 nm). Bismuth is a particularly favorable material to study the electronic properties of quantum wires due to its small electron effective mass and high carrier mobility. We report here the observation of quantum oscillations of the transverse magnetoresistance (MR) in single-crystal Bi nanowires with a diameter d < 80 nm. The nanowire samples with glass coating were prepared by an improved Ulitovsky technique; they were cylindrical single crystals with the (1011) orientation along the wire axis. Nonmonotonic changes of transverse MR that are equidistant in the direct magnetic field were observed at low temperatures in a wide range of magnetic fields up to 14 T. The period of oscillations ΔB ≈ Φ0/S=(h/e)/(πd2/4) depends on the wire diameter d as for the case of longitudinal MR. There are equidistant oscillations of MR on a direct magnetic field under conditions where the magnetic flux through the cylinder Φ = 0. For low magnetic fields, the modulation, which is 10 kΩ, is comparable to the characteristic quantum resistance h/e2 ≈ 25.8 kΩ. The period of oscillations in a wide range of angles (~50o) keeps a constant value. The amplitude of oscillations depends on the angle θ between the direction of applied magnetic field B and the C3 nanowire axis, increasing in intensity with increasing angle θ. At angles θ > 50o the nonmonotonic changes of transverse MR lose their periodicity in a direct magnetic field. Earlier [2, 3], we have observed the oscillations of longitudinal MR of Bi nanowires with two periods proportional to h/e and h/2e. From B ~ 8 T down to B = 0 the extrema of the h/2e oscillation was shifted up to 3π at B = 0, which was the manifestation of the Berry phase shift due to electron moving in the nonuniform magnetic field B B BSO    = + Σ , where BSO is the Zeeman-like effective magnetic field. The fact that the h/2e oscillations exhibit a phase shift only in one direction means that, like in topological insulators, the surface states of Bi nanowires have only one spin degree of freedom. It was recently suggested that bismuth bilayers may manifest quantum spin Hall effect [4]. The Bi crystal can be viewed as a stacking of bilayers with honeycomblike lattice structure along the [111] direction. According to calculation of Murakami [4] Z2 topological number I is odd and Bi bilayer system is inversion symmetric. Probably, transverse MR oscillations arise due to the topological nature of Bi nanowire surface states. This work was partially supported by STCU Grant no. 5373 and SCOPES Grant no. IZ73ZO_127968