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This is a greatly revised and expanded version of the book Cardinal fu- tionsonBooleanalgebras,Birkh. auser1990.Knownmistakesinthatbookhave beencorrected,andmanyoftheproblemsstatedtherehavesolutionsinthepresent treatment. At the same time, many new problems are formulated here; some as development of the solved problems from the earlier work, but most as a result of more careful study of the notions. The book is supposed to be self-contained, and for that reason many classical results are included. ForhelponthisbookIwishtothankE.K.vanDouwen,K.Grant,L. Heindorf, I. Juh' asz, S. Koppelberg, P. Koszmider, P. Nyikos, D. Peterson, M. Rubin, S. Shelah, and S. Todor? cevi' c. Unpublished results of some of these people are contained here, sometimes with proofs, with their permission. As the reader will see, my greatest debt is to Saharon Shelah, who has worked on, and solved, many of the problems stated in the 1990 book as well as in preliminary versions of this book. Of course I am always eager to hear about solutions of problems, mistakes, etc. Electronic lists of errata and the status of the open problems are maintained, initially on the anonymous ftp server of euclid.colorado.edu, directory pub/babib; on www, go to ftp://euclid. colorado.edu/pub/babib.