In this talk I propose firstly to' deal with those aspects of the study of continued fractions which are of direct application in the theory of approximation, and secondly to sketch some developments which are of recent origin and as yet incompletely worked out. It has taken more than two hundred years for the theory of continued fractions to attain its present condition, and I do not propose to speak for quite that length of time. For this reason I shall confine myself to sketching the strategic principles involved in the development of the subject, without going into too much tactical detail.