  We present calculations of the structure functions F_2(x,Q^2) and F_L(x,Q^2),
concentrating on small x. After discussing the standard expansion of the
structure functions in powers of \alpha_s(Q^2) we consider a leading-order
expansion in ln(1/x) and finally an expansion which is leading order in both
ln(1/x) and \alpha_s(Q^2), and which we argue is the only really correct
expansion scheme. Ordering the calculation in a renormalization-scheme-
consistent manner, there is no factorization scheme dependence, as there should
not be in calculations of physical quantities. The calculational method
naturally leads to the ``physical anomalous dimensions'' of Catani, but imposes
stronger constraints than just the use of these effective anomalous dimensions.
In particular, a relationship between the small-x forms of the inputs
F_2(x,Q_0^2) and F_L(x,Q_0^2) is predicted. Analysis of a wide range of data
for F_2(x,Q^2) is performed, and a very good global fit obtained, particularly
for data at small x. The fit allows a prediction for F_L(x,Q^2) to be produced,
which is smaller than those produced by the usual NLO-in-\alpha_s(Q^2) fits to
F_2(x,Q^2) and different in shape.
