  This work is devoted to the so-called filtration theory of semigroup
generators in the unit disk. It should be noted that numerous filtrations
studied to nowdays have been introduced for different purposes and considered
from different points of view. So, our aim is to summarize the known facts, to
present common and distinct properties of filtrations as well as to study some
new filtrations with an emphasis on their connection with geometric function
theory and the dynamic features of semigroups generated by elements of
different filtration families.
  Among the dynamic properties, we mention the uniform convergence on the unit
disk and the sectorial analytical extension of semigroups with respect to their
parameter.
  We also solve the Fekete--Szeg\"o problem over various filtration classes, as
well as over non-linear resolvents.
