Faltings' delta-invariant of compact and connected Riemann surfaces plays a crucial role in Arakelov theory of arithmetic surfaces. In particular, it appears in the arithmetic Noether-formula. We will give new formulas for delta in terms of integrals of theta functions. This has several applications: We obtain an explicit lower bound for delta only depending on the genus and an upper bound for the Arakelov-Green function in terms of delta. Furthermore, we get a canonical extension of delta to the moduli space of indecomposable PPAVs.