  Theoretical expectations concerning small $x$ behaviour of the spin dependent
structure function $g_1$ are summarised. This includes discussion of the Regge
pole model predictions and of the double $ln^2(1/x)$ effects implied by
perturbative QCD. The quantitative implementation of the latter is described
within the unified scheme incorporating both Altarelli-Parisi evolution and the
double $ln^2(1/x)$ resummation. The double $ln^2(1/x)$ effects are found to be
important in the region of $x$ which can possibly be probed at HERA.
Predictions for the polarized gluon distribution $\Delta G(x,Q^2)$ at low $x$
are also given.
