  In this paper we prove new explicit formulas for Faltings' $\delta$-invariant
of an arbitrary hyperelliptic Riemann surface. This has several applications:
For example we obtain an explicit lower bound for $\delta$ depending only on
the genus, and we deduce new explicit bounds for the Arakelov self-intersection
number $\omega^2$ associated to hyperelliptic curves over number fields.
Furthermore, we obtain an improved version of Szpiro's small points conjecture
for hyperelliptic curves of genus at least $3$. Our method allows us in
addition to establish a generalization of Rosenhain's formula on
$\theta$-derivatives conjectured by Gu\`ardia.
