We introduce the idea of F-algebroids as an analogue of Lie algebroids by using F-manifolds introduced by Hertling and Manin. In this talk we will discuss the motivations behind the definition of F-algebroids and will present a characterization of the almost dualty for Frobenius manifolds studied by Dubrovin in terms of F-algebroids. If time permits we will formulate some questions for further developments. This is based on joint work (partially in progress) with Alexander Torres.