These notes are an exposition of Galois Theory from the original Lagrangian and Galoisian point of view. A particular effort was made here to better understand the connection between Lagrange's purely combinatorial approach and Galois algebraic extensions of the latter. Moreover, stimulated by the necessities of present day computer explorations, the algorithmic approach has been given priority here over every other aspect of presentation. In particular, you may not find here the clean simplistic look characteristic of the classical exposition of E. Artin. In contrast these notes should provide a good starting point in attempting constructions in this most difficult computational arena.