Torsions in cohomology groups of manifolds are useful in enormous ways, though they are often harder to control than de Rham  
cohomology. In this talk we'll explain how Artin and Mumford constructed a smooth proper variety over $\mathbb{C}$ with non-trivial  
2-torsion in its $H^3$ and use it to exhibit that unirationality is weaker than stable rationality. If time permits we'll explain why  
torsions in cohomology could be of interest to number theorists.