As an example of a specific instance of the anabelian philosophy, we have Grothendieck's celebrated 'section conjecture', which states (in one form) that for a 'nice' curve $X$ over a number field $F$, the rational points are in bijection with the sections of the exact sequence $$1 \rightarrow \pi_1(X_{\bar{F}}) \rightarrow \pi_1(X) \rightarrow G_F \rightarrow 1$$ where $G_F$ is the absolute Galois group of $F$ and $\pi_1$ is the algebraic (etale) fundamental group. In case the curve is over the complex numbers, the etale $\pi_1$ is the profinite completion of the regular fundamental group, so there is a very close connection to the classical stuff of Hatcher. The conjecture is still a wide open problem, but any proof would mean you could check something of number theoretic interest (existence of rational points on curves) by studying maps between certain generalized homotopy groups!