  We analyze spin-up/spin-down of the neutron star in Be X-ray binary system
GX\,304-1 observed by \textit{Swift}/XRT and \textit{Fermi}/GBM instruments in
the period of the source activity from April 2010 to January 2013 and discuss
possible mechanisms of angular momentum transfer to/from the neutron star. We
argue that the neutron star spin-down at quiescent states of the source with an
X-ray luminosity of $L_x\sim 10^{34}$~erg s$^{-1}$ between a series of Type I
outbursts and spin-up during the outbursts can be explained by quasi-spherical
settling accretion onto the neutron star. The outbursts occur near the neutron
star periastron passages where the density is enhanced due to the presence of
an equatorial Be-disc tilted to the orbital plane. We also propose an
explanation to the counterintuitive smaller spin-up rate observed at higher
luminosity in a double-peak Type I outburst due to lower value of the specific
angular momentum of matter captured from the quasi-spherical wind from the
Be-star by the neutron star moving in an elliptical orbit with eccentricity
$e\gtrsim 0.5$.
