Emerging technologies allow for the creation of nonrectangular displays with unlimited constraints in shape. However, the introduction of such displays radically deviates from the prevailing tradition of placing content on rectangular screens and raises fundamental design questions. Among these is the foremost question of how to legibly present text. We address this fundamental concern through a multi-part exploration that includes: (1) a focus group study from which we collected free-form display scenarios and extracted display shape properties; (2) a framework that identifies different mappings of text onto a non-rectangular shape and formulates hypotheses concerning legibility for different display shape properties; and (3) a series of quantitative text legibility studies to assess our hypotheses. Or results agree with and extend upon other findings in the existing literature on text legibility, but they also uncover unique instances in which different rules need to be applied for non-rectangular displays. These results also provide guidelines for the design of visual interfaces.