I have been asked to write a short account of the present-day work in the algebra of invariants for the benefit of those who have not made a special study of the subject. A good summary account can be found in the various continental works of reference, and particularly in the Encyklopädie der Mathematischen Wissenschaften, Band III. Teil iv. 7, written by Dr. R. Weitzenböck, and brought up to about tho date 1920 with tolerable completeness, both in subject-matter and in references. Nothing has, however, been published in the last thirty years quite so good and readable as Meyer’s Bericht, which reviewed the first fifty years of progress from the time when Boole inaugurated the theory in the Cambridge Mathematical Journal, iii. pp. 1-20, of the year 1841. As things have turned out, and especially because of the impossibility of buying separate parts of the Encyklopädie without embarking on the whole section (Geometry in this particular case), it seems a pity that we have no machinery in England for the systematic issue of corresponding reviews, perhaps in the form of this Bericht, to cover by degrees the field of present-day mathematics.