Long, filamentary structures which are disordered, knotted or tangled are ubiquitous in Nature. Examples range from long molecules and DNA, to magnetic field lines in astrophysical plasmas, to vortex structures or patticle paths in turbulent flows. The aim of this talk is to show that simple ideas from knot theory can be used to characterize the complexity of these structures. The two examples which I shall discuss are turbulence in superfluid helium and classical turbulence in an ordinary viscous fluid. The former is a particularly nice benchmark to apply these ideas, because it consists of a tangle of discrete vortex lines.