We consider Hilbert spaces $H$ which
consist of analytic functions in a domain $\Omega\subset\mathbb{C}$
and have the property that any zero of an element of $H$ which is
not a common zero of the whole space, can be divided out without
leaving $H$. This property is called {\it near invariance} and is
related to a number of interesting problems that connect complex
analysis and operator theory. The concept probably appeared first in
L. de Branges' work on Hilbert spaces of entire functions and played
later a decisive role in the description of invariant subspaces of
the shift operator on Hardy spaces over multiply connected domains.
There are a number of structure theorems for nearly invariant spaces
obtained by de Branges, Hitt and Sarason, and more recently by
Feldman, Ross and myself, but the emphasis of my talk will be on
some applications; the study of differentiation invariant subspaces
of $C^\infty(\mathbb{R})$, or invariant subspaces of Volterra
operators on spaces of power series on the unit disc. Finally, we
discuss near invariance in the vector-valued case and show how it
can be related to kernels of products of Toeplitz operators. More
precisely, I will present in more detail the solution of the
following problem: If a finite product of Toeplitz operators is the
zero operator then one of the factors is zero.