In this talk, we introduce higher-order expansions in the mixing local limit theorems for Birkhoff sums. We will discuss the general results under technical assumptions, and illustrate them by different examples (e.g. subshifts of finite type, Young towers, Sinai billiards, and random matrix products), including situations of unbounded observables with integrability order arbitrarily close to the optimal moment condition required in the i.i.d. setting. This is joint work with Françoise Pène.