  For a smooth projective family of schemes we prove that a semiorthogonal
decomposition of the bounded derived category of coherent sheaves of the fiber
of a point uniquely deforms over an \'etale neighborhood of the point. We do
this using a comparison theorem between semiorthogonal decompositions and
decomposition triangles of the structure sheaf of the diagonal. We then apply
to the latter a deformation theory for morphisms with a fixed lift of the
target, which is developed in the appendix.
  Using this as a key ingredient we introduce a moduli space which classifies
semiorthogonal decompositions of the category of perfect complexes of a smooth
projective family. We show that this is an \'etale algebraic space over the
base scheme of the family, which can be non-quasicompact and non-separated.
This is done using Artin's criterion for a functor to be an \'etale algebraic
space over the base.
  We generalize this to families of geometric noncommutative schemes in the
sense of Orlov. We also define the open subspace classifying non-trivial
semiorthogonal decompositions only, which is used in a companion paper to study
indecomposability of derived categories in various examples.
