In a striking contrast with the classical case of real-valued maps, the space of smooth maps with values into a manifold $N$ need not be dense in the space of $N$-valued $W^{s,p}$ maps. The problem of determining for which values of $s$ and $p$ density occurs, known as the strong density problem, is tied to the topology of the target manifold $N$.

In this talk, I will present in detail the strong density problem, I will review the history of the contributions to its study, and I will explain the complete answer to the problem, with a unified proof covering all interesting values of $s$ and $p$.