In this talk I will review recent results, obtained in collaboration with A. Enciso, on the existence of knotted and linked thin vortex tubes for steady solutions to the incompressible Euler equation. More precisely, given a finite collection of (possibly linked and knotted) disjoint thin tubes, we will see that they can be transformed with a $\mathbb{C}^m$-small diffeomorphism into a set of vortex tubes of a steady solution to the Euler equation that tends to zero at infinity. In particular, we will recover and improve our previous theorem on the existence of knotted and linked vortex lines (Ann. of Math. 175 (2012) 345-367). The proof combines fine energy estimates for a boundary value problem associated to the curl operator with KAM theory and a Runge-type global approximation theorem. The problem of the existence of knotted and linked thin vortex tubes can be traced back to Lord Kelvin, and in fact these structures have been recently realized experimentally by Irvine and Kleckner at Chicago (Nature Phys. 9 (2013) 253-258).