Abstract: The study of "The local structures of Poisson manifolds" was initiated by Alan Weinstein. Among the first results in his 1983 is the splitting theorem, which reduces the problem to classifying Poisson structures around singularities. The case of singularities with semisimple Lie algebra was studied the most. I will mention the classical results by Weinstein, Conn, Monnier and Zung, and the more recent result obtained in collaboration with Florian Zeiser, for isotropy Lie algebra so(3,1)=sl(2,C). If time permits, I will discuss the linearization around leaves (joint work with Marius Crainic) and invariant submanifolds (joint work with Rui Loja Fernandes).