Even during visual fixation of a stationary target object, our eyes perform miniature movements to enhance visual perception. In the first part of my presentation, I will review behavioral findings on fixational eye movements and their function for visual perception. Fixational eye movements show positively correlated increments (i.e., persistence) on a short time scale and negatively correlated increments (i.e., antipersistence) on a longer time scale. Microsaccades (or "flicks") are rapid involuntary movements embedded in slower components of fixational eye movements. Microsaccades serve important visual functions and are triggered by low retinal image slip. In the second part of my presentation, I will discuss first attempts to develop mathematical models of fixational eye movements and microsaccades. I will discuss the physiological plausibility of various theoretical concepts for the control of fixational eye movements, e.g., self-avoiding random walks and time-delayed control of eye velocity. The model can explain and integrate various statistical properties of fixational eye movements and microsaccades.