The evolution of our attempts to model eye movements while people read text is described. All of our models have the core assumption that the primary "engine" that drives the eyes forward in text is the serial identification of words in text. However, the relationship between the identification of a word and the forward movement of attention and the program to fixate the subsequent word is not completely straightforward. The chapter is divided into four major sections. In the first, we describe the "design principles" that guide our modeling. In the second, we summarize our prior published modeling work (Reichle, Pollatsek, Rayner Fisher, 1998; Reichle, Rayner & Pollatsek, 2002). The third section describes our recent efforts to modify our model, both to improve the fit to a corpus of data in English and to make the assumptions more psychologically reasonable. The final section expands our modeling efforts to explain the fixation patterns on Finnish compound words in text, allowing the model to be responsive to constituents of words as well as whole words.