In computability theory, an undecidable problem is a decision problem for which an effective method (algorithm) to derive the correct answer does not exist.
More formally, an undecidable problem is a problem whose language is not a recursive set; see the article Decidable language.
There are uncountably many undecidable problems, so the list below is necessarily incomplete.
Many, if not most, undecidable problems in mathematics can be posed as word problems: determining when two distinct strings of symbols (encoding some mathematical concept or object) represent the same object or not.
For undecidability in axiomatic mathematics, see List of statements undecidable in ZFC.