This lecture starts from two famous discrete-time dynamic models in economics, namely the Cobweb model to describe price dynamics and the Cournot duopoly model to describe competition between two firms producing homogeneous goods, and shows how their study has stimulated new fruitful streams of literature rooted in the field of qualitative analysis of nonlinear discrete dynamical systems. In the case of the Cobweb model, starting from the standard one-dimensional dynamic model the introduction of new kinds of expectations and learning mechanisms open new mathematical research about two-dimensional maps with a vanishing denominator, leading to the study of new kinds of singularities called focal points and prefocal curves. Analogously, in the case of the two-dimensional Cournot duopoly model, some recent developments are described concerning the introduction of nonlinearities leading to multistability, i.e. the coexistence of several stable equilibria, with the related problem of the delimitation of basins of attraction, which requires a global dynamical analysis based on the method of critical curves. Moreover, in the particular case of identical players, some recent results about chaos synchronization and related bifurcations (such as riddling or blowout bifurcation) are illustrated, with extensive reference to the rich and flourishing recent stream of literature.