It is known that the Hamiltonian cycle problem can be solved in a minor-closed class X in polynomial time if and only if X excludes a planar graph, or equivalently, if and only if the tree-width is bounded in X (unless \(P=NP\) ). The family of monotone classes of graphs extends the family of minor-classes, and the polynomial-time solvability of the problem in this family extends far beyond graphs of bounded tree-width. However, the complexity of the problem in the family of monotone classes remains a terra incognita even in the case of a single forbidden subgraph. In the present paper, we identify several crucial landmarks in this territory, introduce the notion of even graphs that plays the fundamental role in the study of the Hamiltonian cycle problem, and prove a number of breakthrough results of both positive and negative nature.