Many results about statistical convergence in a general setting can be derived by consideration of the convergence free space of null sequences and its related cofilter and filter. We derive algebraic and order properties, preservation under uniform convergence, Cauchy properties, and properties of cluster points. We also show that for certain types of statistical convergence stronger convergence properties also hold. Finally we discuss the relationship between generalized statistical convergence and subsets of the Stone-Cech compactification of the integers. (C) 2000 Academic Press.