Accepted Friday Jul 24, 2009
Progress over the last two decades in the theory of crystal surfaces in and out of equilibrium is reviewed. Various instabilities that occur during growth, sublimation, or are caused by elasticity, electromigration, etc... are addressed. For several geometries and nonequilibrium circumstances, a systematic derivation provides various continuum nonlinear evolution equations for driven stepped (or vicinal) surfaces. The resulting equations are sometimes different from the phenomenological equations derived from symmetry arguments, such as Kardar-Parisi-Zhang's. Some of the evolution equations are met in other nonlinear dissipative systems, while others remain unrevealed. The novel and original class of equations are referred to as "non-standard". This non-standard form suggests nontrivial dynamics, where phenomenology and symmetries, often used to infer evolution equations, fail to produce the correct form. This review focuses on step meandering and bunching, which are the two main forms of instabilities encountered on vicinal surfaces. Standard and non-standard evolution scenarios are presented using a combination of physical arguments, symmetries and systematic analysis. Other features like kinematic waves, some aspect of nucleation, and results of kinetic Monte Carlo simulations are also presented. The current state of experiments and confrontation with theories are discussed. Challenging open issues raised by the recent progress, which constitute essential future lines of inquiries, are outlined.