After the pioneering works of Kohn ('63, '64), Morrey ('58) and Hormander ('65) in which the $L^2$-existence and the $C^\infty$ regularity in the interior of the solutions of the $\bar\partial$-equation in pseudoconvex domains were proved, many mathematicians worked to extend these results for strictly q-convex domains (Fischer, Lieb '74; Henkin, Leiterer '88, Schmalz '89, etc ...). The problem of local boundary regularity in strictly q-convex domains was solved by Kohn ('65)  who even proved that $\frac 12$-subelliptic estimates hold. In 1979 Kohn gave a geometrically characterization of real analytic pseudoconvex domains for which $\epsilon$-subelliptic estimates hold ($0<\epsilon<\frac 12$). In a paper of 1991 Ho generalized some results of Kohn for weakly $q$-convex domains, i.e. the sum of the first q eigenvalues of the Levi form at the boundary is non-negative. In this talk I will explain how to generalize the $L^2$-existence and the $C^\infty$ regularity in the interior when the boundary is non-smooth and also I will give a local geometric characterization of a real analytic q-convex domains for which $\epsilon$-subelliptic estimates hold. These are joint works with Fassina (2018) and  Zampieri (2015).