In this talk we present a survey of our results on quasi-periodic Jacobi operators whose diagonal and off-diagonal elements are generated from two analytic functions on the circle. Such operators arise as effective Hamiltonians describing the effects of external magnetic fields on a tight binding, infinite crystal layer. The main motivation of our investigations was extended Harper's model (EHM), whose description on both the level of spectral analysis, as well the Lyapunov exponent (LE) had posed an open problem even from the point of view of physics literature. Among the topics that will be addressed are: Singular components of spectral measures for ergodic Jacobi operators, Singular analytic cocycles and joint continuity of the Lyapunov exponent, Recovery of spectral data from rational frequency approximants, Almost constant cocycles and the complexified LE of EHM, Spectral theory of EHM.