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I shall describe some results and some open problems concerning dense ideals in unital topological algebras (mostly in commutative $F$-algebras, i.e. complete metric algebras). The following topics will be considered:
(i) When the algebra $A$ in question has a dense ideal.
(ii) When there are dense ideals of innite co-dimension (as e.g. in the algebra $\varepsilon$ of all entire functions with the compact-open topology).
(iii) When $A$ has a non-closed radical.
(iv) When $A$ has proper (i.e. non-invertible) topologically invertible elements.