In this paper, we argue that deriving the nature of circumstellar media (CSM) around type Ia supernovae (SNe Ia) is key to understanding their progenitor system(s). The issue is still controversial for normal SNe Ia, while there is robust evidence of dense CSM for a few classes of SN Ia `outliers'. We discuss the nature of CSM and the inferred progenitor systems for the candidate super-Chandrasekhar SNe Ia (SC-SNe Ia; SNe Ia originally characterized by large luminosity) and for the so-called SNe Ia-CSM (SNe Ia showing signatures of a strong ejecta-CSM interaction). The nature of CSM around SC-SN Ia 2012dn seems largely consistent with the expectation from an extreme case of the single degenerate scenario. The CSM around SNe Ia-CSM requires a very intensive mass loss rate, which may indicate that an Asymptotic Giant Branch (AGB) star would be involved in their progenitor systems. The link between SN Ia-CSM and SN 1991T-like objects is intriguing, and there is a possibility that they would arise from the same progenitor system while the difference is due to a variance in the progenitor configuration.