I discuss how a Tauberian theorem for singular values of noncommuting operators allows us to prove Weyl asymptotic formulas in noncommutative geometry at a high level of generality. This can be applied to obtain far-reaching extensions of Connes’ integration formula in a noncommutative setting. We explain how, via the Birman-Schwinger principle, these asymptotics imply that a semiclassical Weyl law holds for many interesting noncommutative examples.