  We introduce a concept of approximately invertible elements in non-unital
normed algebras which is, on one side, a natural generalization of
invertibility when having approximate identities at hand, and, on the other
side, it is a direct extension of topological invertibility to non-unital
algebras. Basic observations relate approximate invertibility with concepts of
topological divisors of zero and density of (modular) ideals. We exemplify
approximate invertibility in the group algebra, Wiener algebras, and operator
ideals. For Wiener algebras with approximate identities (in particular, for the
Fourier image of the convolution algebra), the approximate invertibility of an
algebra element is equivalent to the property that it does not vanish. We also
study approximate invertibility and its deeper connection with the Gelfand and
representation theory in non-unital abelian Banach algebras as well as abelian
and non-abelian C*-algebras.
