In 1943 R. C. Buck showed that a sequence x is convergent if some regular matrix sums every subsequence of x. Thus, for example, if every subsequence of x is Cesaro summable, then x is actually convergent. Buck's result was quite surprising, since research in summability theory up to that time gave no hint of such a remarkable theorem. The appearance of Buck's result in the Bulletin of the American Mathematical Society (3) created immediate interest and has prompted considerable research which has taken the following directions: (i) to study regular matrix transformations in order to shed light on Buck's theorem, (ii) to extend Buck's theorem, (iii) to obtain analogs of Buck's theorem for sequence spaces other than the space of convergent sequences, and (iv) to obtain analogs of Buck's theorem involving processes other than subsequencing, such as stretching. The purpose of the present paper is to contribute to all facets of the problem, particularly to (i), (iii), and (iv).