In characteristic 0 irreducible tensor products of representations of symmetric and alternating groups have been described by Zisser and by Bessenrodt and Kleshchev. In positive characteristic a classification conjecture for such products for symmetric groups was formulated by Gow and Kleshchev. Parts of the conjecture were proved shortly after in papers of Bessenrodt and Kleshchev and of Graham and James. However many cases in characteristic 2 for n even were still open. For alternating groups in characteristics p at least irreducible tensor products have been described in a paper of Bessenrodt and Kleshchev, though not for smaller p.In the submitted papers I consider the still open cases, completing the classification of irreducible tensor products of representations of symmetric and alternating groups up to a certain class of tensor products for alternating groups in characteristic 2.