Poisson manifolds appear as general phase spaces in classical mechanics. A Poisson manifold is called symplectic, or nondegenerate, if locally it has coordinates satisfying the standard canonical relations in Hamiltonian mechanics. The idea of realizing a Poisson bracket by nondegenerate or symplectic structures can be traced back to S. Lie in the 19th century. The existence of symplectic realizations for arbitrary Poisson manifolds was proved independently by Karasev and Weinstein in the late 1980s. In this talk I will discuss some recent developments around this topic. In particular, I will explain a theorem by Mackenzie and myself about the integration of Lie bialgebroids. Our theorem not only gave a new proof for the Karasev and Weinstein theorem as a consequence, but also solved some mystery surrounding their theorem about an additional structure of the so-called symplectic groupoids. Such a structure of symplectic groupoids is also related to the Kontsevich *-products, as recently shown by Cattaneo and Felder.