This paper proposes some simple models where the central bank trades off between stabilizing the business cycle and targeting inflation to a level that stabilizes the public debt ratio. We show that in a closed economy fiscal dominance does not necessarily imply hyperinflation. Moreover, in an open economy it is successful in lowering debt ratios when output is reactive enough to unconventional monetary policy and when the expectations of future inflation are well anchored to the debt-stabilization inflation target. We show that the dynamics of both inflation and public debt ratio are described by first-difference equations with time varying coefficients. We provide some conditions for the asymptotic solutions of the long-run steady states. In particular, we define two regimes of respectively strong and weak fiscal dominance, depending upon whether or not the central bank's action ensures both the sustainability and the speed of convergence of debt to its long-term level.