Parton distributions at small<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>x</mml:mi></mml:math>

We perform a next-to-leading-order QCD analysis of the recent data for deep-inelastic lepton-nucleon scattering and related processes, in which we pay particular attention to the forms of the parton distributions at very small $x$. We discuss in detail, and we incorporate in the analysis, the theoretical QCD results leading to the singular ${x}^{\frac{\ensuremath{-}1}{2}}$-type behavior of the gluon and sea-quark distributions, as well as the modifications due to shadowing effects. We find the QCD shadowing corrections are significant for $x\ensuremath{\lesssim}{10}^{\ensuremath{-}3}$ even though the parton distributions are below their saturation limit. We give predictions for the structure functions ${F}_{2}$ and ${F}_{L}$ accessible at the DESY $\mathrm{ep}$ collider HERA, and for $W$ and $Z$ production up to the energies of the CERN Large Hadron Collider and the Superconducting Super Collider. We discuss the possibility of experiments at these colliders probing the parton distributions in the very-small-$x$ region.