  In this paper we explore a recently emerged approach to the problem of
quantisation based on the notion of quantisation ideals. We explicitly prove
that the nonabelian Volterra together with the whole hierarchy of its
symmetries admit a deformation quantisation. We show that all odd-degree
symmetries of the Volterra hierarchy admit also a non-deformation quantisation.
We discuss the quantisation problem for periodic Volterra hierarchy including
their quantum Hamiltonians, central elements of the quantised algebras, and
demonstrate super-integrability of the quantum systems obtained. We show that
the Volterra system with period $3$ admits a bi-quantum structure, which can be
regarded as a quantum deformation of its classical bi-Hamiltonian structure.
