In this talk we present the results on the study of finite-dimensional nilpotent and solvable Poisson algebras. We establish classical results such as Engel’s theorem and Lie’s theorem for Poisson algebras, and examine the role of idempotents in these algebras. The construction of nilpotent and solvable Poisson algebras, exploring the existence of Poisson algebras associated with a fixed Lie algebra, and constructions involving the tensor product and generalized Jacobians will be presented. Furthermore, we show that, under mild restrictions, the solvability and nilpotency of a Poisson algebra are essentially determined by those of the Lie bracket. A criterion for the non-existence of Poisson algebra structures on solvable extensions of nilpotent Lie algebras by a torus will be provided. In particular, we show that complete solvable Lie algebras do not admit a Poisson algebra structure.