  The proton spin puzzle is a longstanding problem in high-energy nuclear
physics: how the proton spin distributes among the spin and orbital angular
momenta of the quarks and gluons inside. Two of the unresolved pieces of the
puzzle are the contributions to quark and gluon spins from the region of small
Bjorken $x$. This dissertation fills the gap by constructing the evolution of
these quantities into the small-$x$ region using a modified dipole formalism.
The dominant contributions to the evolution equations resum powers of
$\alpha_s\ln^2(1/x)$, where $\alpha_s$ is the strong coupling constant. In
general, these evolution equations do not close. However, once the large-$N_c$
or large-$N_c\& N_f$ limit is taken, they turn into a system of linear integral
equations that can be solved iteratively. At large $N_c$, the evolution
equations are shown to be consistent with the gluon sector of the polarized
DGLAP evolution in the small-$x$ limit. We numerically solve the equations in
the large-$N_c$ and large-$N_c\& N_f$ limits and obtain the exponential growth
in $\ln(1/x)$ for $N_f \leq 5$, with the intercept decreasing with $N_f$. For
the large-$N_c$ limit, we have $\alpha_h = 3.66$, which agrees up to the
uncertainty with the earlier work by Bartels, Ermolaev and Ryskin. Furthermore,
at $N_f=6$, the asymptotic form attains an oscillation in $\ln(1/x)$ on top of
the exponential growth, with the period spanning several units of rapidity.
Finally, parts of the single-logarithmic corrections to the small-$x$ helicity
evolution is also derived, resumming powers of $\alpha_s\ln(1/x)$. There, the
effects of the unpolarized small-$x$ evolution and the running coupling are
also included for consistency. The complete single-logarithmic corrections can
be derived based on the framework established here. Altogether, these equations
will provide the most precise small-$x$ helicity evolution to date.
