Our aim in this survey is to present a general view (in a very personal approach) of the main advances in non-commutative algebra during the twentieth century, mainly in its second half. In the first part, we focuss on problems that were solved in the last half of the XX Century, but were originated long ago and they are nucleated around the restricted Burnside problem. ln the second part emphasis is given to very new problems, generated and solved in the last part of the century. In this case, the exposition is articulated around the Moonshine Conjecture.