In the present paper, we describe the recent approach to residue currents by M. Andersson, J. E. Björk, H. Samuelsson, focusing primarily on the methods inspired by analytic continuation. Coleff-Herrera currents (with or without poles) play indeed a crucial role in Lelong-Poincaré type factorization formulas for integration currents on reduced closed analytic sets. As revealed by local structure theorems (which could also be understood as global when working on a complete algebraic manifold due to the GAGA principle), such objects are of algebraic nature (antiholomorphic coordinates playing basically the role of ''inert'' constants). Thinking about division or duality problems instead of intersection ones (especially in the ''improper'' setting, which is certainly the most interesting), it happens then to be necessary to revisit from this point of view the multiplicative inductive procedure initiated by N. Coleff and M. Herrera}, this being the main objective of this presentation. In hommage to the pioneer work of Leon Ehrenpreis, to whom we are both deeply indepted, and as a tribute to him, we also suggest a currential approach to the so-called Noetherian operators, which remain the key stone in various formulations of Leon's Fundamental Principle.Read less <