  We study the well-posedness of nonautonomous nonlinear delay equations in
$\mathbb{R}^{n}$ as evolutionary equations in a proper Hilbert space. We
present a construction of solving operators (nonautonomous case) or nonlinear
semigroups (autonomous case) for a large class of such equations. The main idea
can be easily extended for certain PDEs with delay. Our approach has lesser
limitations and much more elementary than some previously known constructions
of such semigroups and solving operators based on the theory of accretive
operators. In the autonomous case we also study differentiability properties of
these semigroups in order to apply various dimension estimates using the
Hilbert space geometry. However, obtaining effective dimension estimates for
delay equations is a nontrivial problem and we explain it by means of a scalar
delay equation. We also discuss our adjacent results concerned with inertial
manifolds and their construction for delay equations.
