Abstract: The "Weyl fermion" was discovered in a topological semimetal in 2015. Its mathematical characterisation turns out to involve deep and subtle results in differential topology. I will outline this theory, and explain some connections to Euler structures, torsion of manifolds, and Seiberg-Witten invariants. I also propose interesting generalisations with torsion topological charges arising from Kervaire semicharacteristics and "Quaternionic" characteristic classes.