The functional notion of "topic" or "topicality" has suffered, traditionally, from two distinct drawbacks. First, it has remained largely ill defined or intuitively defined. And second, quite often its definition boiled down to structure-dependent circularity. This volume represents a major departure from past practices, without rejecting both their intuitive appeal and the many good results yielded by them. First, "topic" and "topicality" are re-analyzed as a scalar property, rather than as an either/or discrete prime. Second, the graded property of "topicality" is firmly connected with sensible cognitive notions culled from gestalt psychology, such as "predictability" or "continuity". Third, we develop and utilize precise measures and quantified methods by which the property of "topicality" of clausal arguments can be studied in connected discourse, and thus be properly hinged in its rightful context, that of topic identification, maintenance and recoverability in discourse. Fourth, we show that many grammatical phenomena which used to be studied by linguists in isolation, all partake in one functional domain of grammar, that of topic identification. Finally, we demonstrate the validity of this new approach to the study of "topic" and "topicality" by applying the same text-based quantifying method to a number of typologically-diverse languages, in studying actual texts. Languages studied here are: Written and spoken English, spoken Spanish, Biblical Hebrew, Amharic, Hausa, Japanese, Chamorro and Ute.