  This is the first in a series of papers dedicated to the study of Poisson
manifolds of compact types (PMCTs). This notion encompasses several classes of
Poisson manifolds defined via properties of their symplectic integrations. In
this first paper we establish some fundamental properties of PMCTs, which
already show that they are the analogues of compact symplectic manifolds, thus
placing them in a prominent position among all Poisson manifolds. For instance,
their Poisson cohomology behaves very much like the de Rham cohomology of
compact symplectic manifolds (Hodge decomposition, non-degenerate Poincar\'e
duality pairing, etc.) and the Moser trick can be adapted to PMCTs. More
important, we find unexpected connections between PMCTs and Symplectic
Topology: PMCTs are related with the theory of Lagrangian fibrations and we
exhibit a construction of a nontrivial PMCT related to a classical question on
the topology of the orbits of a free symplectic circle action. In subsequent
papers, we will establish deep connections between PMCTs and integral affine
geometry, Hamiltonian $G$-spaces, foliation theory, Lie theory and symplectic
gerbes.
