  We investigate approximation to a given real number by algebraic numbers and
algebraic integers of prescribed degree. We deal with both best and uniform
approximation, and highlight the similarities and differences compared with the
intensely studied problem of approximation by algebraic numbers (and integers)
of bounded degree. We establish the answer to a question of Bugeaud concerning
approximation to transcendental real numbers by quadratic irrational numbers,
and thereby we refine a result of Davenport and Schmidt from 1967. We also
obtain several new characterizations of Liouville numbers, and certain new
insights on inhomogeneous Diophantine approximation. As an auxiliary side
result, we provide an upper bound for the number of certain linear combinations
of two given relatively prime integer polynomials with a linear factor. We
conclude with several open problems.
