  In this paper, we study deformations of compact holomorphic Poisson
submanifolds which extend Kodaira's series of papers on semi-regularity
(deformations of compact complex submanifolds of codimension 1), deformations
of compact complex submanifolds of arbitrary codimensions, and stability of
compact complex submanifolds in the context of holomorphic Poisson
deformations. We also study simultaneous deformations of holomorphic Poisson
structures and holomorphic Poisson submanifolds on a fixed underlying compact
complex manifold. In appendices, we present deformations of Poisson closed
subschemes in the language of functors of Artin rings which is the algebraic
version of deformations of holomorphic Poisson submanifolds. We identify
first-order deformations and obstructions.
