Abstract
The Skyrme model (1961) was one of the first  (and best) attempts to describe elementary particles
as localized in space solutions of nonlinear PDEs. The fields take their values in $SU(2)=S^3$ and
stabilize at spatial infinity. Thus, the configuration space splits into different sectors (homotopy classes)
with a constant (integer) topological charge (the degree) in each sector. Faddeev's model (1975)  was
designed to provide additional internal structure (knottedness) to the localized solutions.The fields
take values in $S^2$ and the topological charge is the Hopf invariant. In this talk I will discuss some old
and new results for these models.