Teichmueller curves are central objects in geometry and dynamics. They provide fertile connections between polygon billiards, flat surfaces and moduli spaces. A class of special Teichmueller curves come from a branched cover construction. Using them as examples, I will introduce an algebro-geometric technique to study Teichmueller curves. As applications, we prove Kontsevich-Zorich's conjecture on the non-varying property of Siegel-Veech constants and the sum of Lyapunov exponents for Abelian differentials in low genus. Moreover, we provide a novel approach to the Schottky problem of describing geometrically the locus of Jacobians among Abelian varieties. This talk will be accessible to a general audience.