Abstract: The class of quasiperiodic operators with unbounded monotone potentials is a natural generalization of the Maryland model. In one dimension, we show that Anderson localization holds at all couplings for a large class of Lipschitz monotone sampling functions. The method is partially based on earlier results joint with S. Jitomirskaya on the bounded monotone case. We also establish that the spectrum is the whole real line. In higher dimensions, we tentatively establish perturbative Anderson localization by showing directly that eigenvalue and eigenfunction perturbation series are convergent. Compared to the previously known KAM localization proof by Bellissard, Lima, and Scoppola, our approach gives explicit diagram-like series for eigenvalues and eigenfunctions, and allows a larger class of potentials. The higher-dimensional results are joint with L. Parnovski and R. Shterenberg.