A large proportion of the Bayesian mechanism design literature is restricted to the family of regular distributions $\mathbb{F}_{\tt reg}$ [Mye81] or the family of monotone hazard rate (MHR) distributions $\mathbb{F}_{\tt MHR}$ [BMP63], which overshadows this beautiful and well-developed theory. We (re-)introduce two generalizations, the family of quasi-regular distributions $\mathbb{F}_{\tt Q-reg}$ and the family of quasi-MHR distributions $\mathbb{F}_{\tt Q-MHR}$. All four families together form the following hierarchy: $\mathbb{F}_{\tt MHR} \subsetneq (\mathbb{F}_{\tt reg} \cap \mathbb{F}_{\tt Q-MHR}) \subsetneq \mathbb{F}_{\tt Q-reg}$ and $\mathbb{F}_{\tt Q-MHR} \subsetneq (\mathbb{F}_{\tt reg} \cup \mathbb{F}_{\tt Q-MHR}) \subsetneq \mathbb{F}_{\tt Q-reg}$. The significance of our new families is manifold. First, their defining conditions are immediate relaxations of the regularity/MHR conditions (i.e., monotonicity of the virtual value functions and/or the hazard rate functions), which reflect economic intuition. Second, they satisfy natural mathematical properties (about order statistics) that are violated by both original families $\mathbb{F}_{\tt reg}$ and $\mathbb{F}_{\tt MHR}$. Third but foremost, numerous results [BK96, HR09a, CD15, DRY15, HR14, AHN+19, JLTX20, JLQ+19b, FLR19, GHZ19b, JLX23, LM24] established before for regular/MHR distributions now can be generalized, with or even without quantitative losses.