  A major direction in the theory of cluster algebras is to construct (quantum)
cluster algebra structures on the (quantized) coordinate rings of various
families of varieties arising in Lie theory. We prove that all algebras in a
very large axiomatically defined class of noncommutative algebras possess
canonical quantum cluster algebra structures. Furthermore, they coincide with
the corresponding upper quantum cluster algebras. We also establish analogs of
these results for a large class of Poisson nilpotent algebras. Many important
families of coordinate rings are subsumed in the class we are covering, which
leads to a broad range of application of the general results to the above
mentioned types of problems. As a consequence, we prove the
Berenstein--Zelevinsky conjecture for the quantized coordinate rings of double
Bruhat cells and construct quantum cluster algebra structures on all quantum
unipotent groups, extending the theorem of Gei\ss, Leclerc and Schr\"oer for
the case of symmetric Kac--Moody groups. Moreover, we prove that the upper
cluster algebras of Berenstein, Fomin and Zelevinsky associated to double
Bruhat cells coincide with the corresponding cluster algebras.
