In this paper we discuss well-known three- and two-dimensional models with confinement, namely, the Polyakov compact electrodynamics in 3D and two-dimensional CP(N-1) sigma model, and reveal changes in the confining regimes of these models upon adding the axion field. In both cases the addition of axion has a drastic impact. In the CP(N-1) model the axion-induced deconfinement was known previously, but we discuss a new feature not considered in the previous publication.