High-resolution measurements of positron interactions with Ne and Ar are presented, as well as theoretical treatments. The data extend over a range of 0.3 to 60 eV and comprise measurements of the grand total, positonium formation, and grand total minus positronium formation cross sections. Theoretical treatments of scattering from Ne and Ar are performed under the relativistic optical potential approach, as well as calculations using the convergent close-coupling method. Comparisons of the present measurements and theories are made with previous theoretical and experimental work.