This work presents new achievements and results concerning duality in scalar convex optimization with applications in the theory of maximal monotone operators. Regularity conditions guaranteeing Fenchel and Lagrange duality expressed by means of generalized interiority notions like the quasi interior and the quasi-relative interior, are investigated. Several necessary and sufficient sequential optimality conditions for a general convex optimization problem are derived, overcoming in this way the situation when no regularity condition is fulfilled. The second part of this work follows the lines of the modern Monotone Operator Theory. Connections between convex optimization (especially conjugate duality) and the theory of enlargements of maximal monotone operators are underlined. The final part of the thesis proposes a study of enlargements of positive sets, a generalization of the notion of enlargement of maximal monotone operators.