  We discuss recent developments in the study of semiorthogonal decompositions
of algebraic varieties with an emphasis on their behaviour in families. First,
we overview new results concerning homological projective duality. Then we
introduce residual categories, discuss their relation to small quantum
cohomology, and compute Serre dimensions of residual categories of complete
intersections. After that we define simultaneous resolutions of singularities
and describe a construction that works in particular for nodal degenerations of
even-dimensional varieties. Finally, we introduce the concept of absorption of
singularities which works under appropriate assumptions for nodal degenerations
of odd-dimensional varieties.
