In this talk I will focus on systems which display certain hyperbolic features. Inspired by the work of Lai-Sang Young '98 on constructing SRB measures, Pesin, Senti, and Zhang '16 introduced maps with inducing schemes of hyperbolic type and effected thermodynamical formalism for such maps, i.e., constructed equilibrium measures for a broad class of potentials. Particular examples of maps admitting inducing schemes of hyperbolic type include: the H\'enon map at first bifurcation, the Katok map, and a slow-down of a classical Smale-Williams solenoid. The aim of this talk is to present a comprehensive view on the subject and to describe some new results regarding ergodic properties of such systems, such as decay of correlations, the Central Limit Theorem, the Bernoulli property, etc. This is a joint work with Farruh Shahidi.