Abstract: Paraproducts can be thought of "parts of a product" of two functions, that isolate particular properties of each of the functions. On the real line, dyadic paraproducts decompose a product of two functions, while in complex analysis, Hankel and Toeplitz operators play the same role. We discuss questions motivated by the study of Toeplitz operators in real analysis, and classify the boundedness of a composition of two dyadic paraproducts. In the second part, motivated by the theory of paraproducts in the real valued setting, we consider the question of two-weight boundedness of Hankel operators, with Muckenhoupt weights. We establish conditions under which a Hankel operator is bounded between two weighted spaces, with possibly different weights.