When ten years ago (in the autumn of 1876) I accepted the enticing invitation of Sir Wyville Thomson to undertake the investigation of these microscopic creatures, I hoped to be able to accomplish the task with some degree of completeness within a period of from three to five years, but the further my investigations proceeded the more immeasurable seemed the range of forms, like the boundless firmament of stars. I soon found myself compelled to decide between making a detailed study of a selection of special forms or giving as complete a survey as possible of the varied forms of the whole class; and I decided upon the latter course, having regard both to the general plan of the Challenger Reports, and to the interests of our acquaintance with the class as a whole. I must, however, confess at the close of my work that my original intention is far from having been fulfilled. The extraordinary extent and varied difficulties of the undertaking must excuse the many deficiencies. The special examination of the Challenger collection was for the most part completed in the summer of 1881; I collected its results in my Entwurf eines Radiolarien-Systems auf Grund von Studien der Challenger-Radiolarien (Jenaische Zeitschr. f. Naturw., Bd. xv., 1881). Since the manuscript of this preliminary communication was completed only a few days before my departure for Ceylon, and since I was unable to correct the proofs myself, several errors have crept into the Prodromus Systematis Radiolarium included in it. These have been corrected in the following more extensive working out of it. Even at that time I had distinguished 630 genera and more than 2000 species; but on the revision of these, which I undertook immediately on my return from India, this number was considerably increased. The total number of forms here described amounts to 739 genera and 4318 species; of these 3508 are new, as against 810 previously described. In spite of this large number, however, and in spite of the astonishing variety of the new and marvellous forms, the riches of the Challenger collection are by no means exhausted. A careful and patient worker who would devote a second decade to the work, would probably increase the number of new forms (especially of the smaller ones) by more than a thousand; but for a really complete examination, the lifetime of one man would not suffice.