  We define theta blocks as products of Jacobi theta functions divided by
powers of the Dedekind eta-function and show that they give a powerful new
method to construct Jacobi forms and Siegel modular forms, with applications
also in lattice theory and algebraic geometry. One of the central questions is
when a theta block defines a Jacobi form. It turns out that this seemingly
simple question is connected to various deep problems in different fields
ranging from Fourier analysis over infinite-dimensional Lie algebras to the
theory of moduli spaces in algebraic geometry. We give several answers to this
question.
