  A fully analytical description of the allowed $\beta$ spectrum shape is given
in view of ongoing and planned measurements. Its study forms an invaluable tool
in the search for physics beyond the standard electroweak model and the weak
magnetism recoil term. Contributions stemming from finite size corrections,
mass effects, and radiative corrections are reviewed. A particular focus is
placed on atomic and chemical effects, where the existing description is
extended and analytically provided. The effects of QCD-induced recoil terms are
discussed, and cross-checks were performed for different theoretical
formalisms. Special attention was given to a comparison of the treatment of
nuclear structure effects in different formalisms. Corrections were derived for
both Fermi and Gamow-Teller transitions, and methods of analytical evaluation
thoroughly discussed. In its integrated form, calculated $f$ values were in
agreement with the most precise numerical results within the aimed for
precision. We stress the need for an accurate evaluation of weak magnetism
contributions, and note the possible significance of the oft-neglected induced
pseudoscalar interaction. Together with improved atomic corrections, we then
present an analytical description of the allowed $\beta$ spectrum shape
accurate to a few parts in $10^{-4}$ down to 1\,keV for low to medium $Z$
nuclei, thereby extending the work by previous authors by nearly an order of
magnitude.
