  We show that every compact complex analytic space endowed with a fine
logarithmic structure and every morphism between such spaces admit a
semi-universal deformation. These results generalize the analogous results in
complex analytic geometry first independently proved by A. Douady and H.
Grauert in the '70. We follow Douady's two steps process approach consisting of
an infinite-dimensional construction of the semi-universal deformation space
followed by a finite-dimensional reduction.
