A consistent approach is given to the stability of random-access systems with feedback based on the fact that such systems are generally well modelled by homogeneous Markov chains. The random-access system is stable or unstable according as the Markov chain is stable (ergodic) or unstable (not ergodic). Markov-chain theory, in particular Foster's theorem and Kaplan's converses thereto, provide conditions for stability and instability. Following this approach, it is shown that non-adaptive ALOHA-type algorithms such as pure ALOHA and slotted ALOHA are always unstable. Two important random-access systems based on conflict-resolution protocols, the basic stack algorithm with blocked access and the frugal stack algorithm with blocked access, are analyzed for stability, first in a rudimentary way, then in a precise manner based on a novel solution of the functional recursion describing the system. The precise analysis, while rigorous, employs only elementary mathematics. A by-product of the analysis herein is the formulation of a condition for a homogeneous Markov chain to be unstable that is well adapted for application in random-access systems.