  The present status of the renormalization-group (RG) predictions on the
superfluid density $\rho_s$ and the specific heat $C^+$ and $C^-$ near the
$\lambda$-line of $^4He$ is briefly reviewed. Particular attention is given to
universal amplitude ratios related to these quantities. The goals of a new
theory project are presented that involves higher-order calculations of the
amplitude functions of $\rho_s, C^+$ and $C^-$ which are of fundamental
importance for a test of the universality predictions of the RG theory.
