The theory of L-splines is a unifying representation of classical splines relying on differential operators. Recently, we focused on new spline models inspired from Hermite interpolation. Hermite splines differ from their classical counterpart by incorporating information about derivatives (possibly of various orders) at each control point location. We have been able to build several Hermite spline families, but are lacking a global theoretical framework that would comprise them all. The goal of this project is to build a general theory of Hermite L-splines, where L is a suitable differential operator. This implies studying the required properties of the operator, show that the existing Hermite spline families fit in the unifying representation, and derive new families. Strong mathematical interest is required, and knowledge on approximation theory and functional analysis is a plus.