Abstract Recent advances and problems in pteridological research. - The first part of this invited lecture deals with the taxonomic position of the Pteridophyta in the Vegetable Kingdom (Chart 1) and particularly the recognition of them as an independent division, distinct from Bryophyta as well as from Spermatophyta but nearer the latter than to the former. Accordingly, in the author's opinion, the Cormobionta can be subdivided into two main groups (Chart 1, and Fig. 2): the Bryophytonta, with the sole division of Bryophyta, and the Stelophytonta, consisting of the Pteridophyta and Spermatophyta. The Bryophytonta are regarded as a conservative phyletic line with low perspective power, running independently since the most remote time from the Stelophytonta (Fig. 1), which, rich in evolutive potentialities, were by far more successful than the former. The extant Pteridophyta are classified according to the scheme proposed by the author in 1977, which shows all taxa from the rank of subdivision to that of family (Chart 2). The second part of the lecture is mainly devoted to the results of some recent cytogenetic investigations on the Pteridophytes. First of all, information is given on the processes by which the diploids give rise to other entities mainly autotetraploids and allotetraploids (Figs. 3 & 4). Recent cytogenetic investigations have greatly contributed to the knowledge of the genesis of many European, Mediterranean, Macaronesian and North American species. This lecture deals with the species of some of these genera. The genus Polypodium is represented in Europe by three species, a diploid, a tetraploid and a hexaploid, the probable interrelationships in them are shown in Fig. 5. The origin of the tetraploid, P. vulgare, is not definitely ascertained, but presumably its parents are two species which do not occur at present in Europe and thus its formation is presumably rather ancient; its centre of origin is uncertain. The genus Cheilanthes (excl. Notholaena and Cosentinia) is represented in the Mediterranean Region, Europe and Macaronesia by seven species: four diploids and three allotetraploids (Fig. 6). Three other allotetraploids would be theoretically possible, but they are so far unknown. Presumably their formation did not take place since their ranges are geographically far apart from each other. The species of Cheilanthes are regarded as relicts of the xerophilous palaeomediterranean flora. In connection with this subject, the scarcity of strongly xerophilous plants among the Pteridophyta and their scattered occurrence in different families and orders is pointed out. Asplenium is circumscribed in a narrow sense, and Ceterach, Phyllitis and Pleurosorus are regarded as independent genera. The interrelationships of diploids, hybrids and tetraploids, both auto- and allotetraploids, are shown in Chart 3 which also gives the genomic formulae of the pertinent hybrids, whose names are given in an independent list. The scheme shows also those tetraploids and hybrids which have been synthesized but are not yet found in nature; while those diploids which produce neither hybrids with the other species and between themselves, nor tetraploid progeny are left out. Some observations of various nature pertinent to several species are made in addition to the scheme. Particular attention is paid to the delayed allopolyploidy. This kind of ploidy, in our case an allotetraploidy, is interpreted as an attempt of some species to give a tetraploid progeny, when the hybridization between their diploids is prevented by ecological or chorological barriers, or by a too scarce affinity. An example of the process of formation of delayed allotetraploidy is given in Fig. 7. The two small satellite genera Ceterach and Phyllitis are able to produce hybrids with Asplenium from which they probably took rise in remote times. One of these intergeneric hybrids is × Asplenoceterach; of which the best known is × A. badense, which is regarded as a delayed allotetraploid (Fig. 7). Other presumed hybrids between these genera are × A. barrancense and × A. newmanii. The latter is not yet cytologically investigated. The intergeneric hybrids between Asplenium and Phyllitis are named × Asplenophyllitis. All of them are triploid and those which have been found in nature are included in a scheme (Fig 8) in which the relationships of their common diploid parent, Phyllitis scolopendrium subsp. scolopendrium with four different tetraploid parents are shown. Two of the satellite genera of Asplenium, namely Ceterach and Phyllitis, hybridize between themselves as well. Perhaps the plant found recently in Corsica and named Asplenium dutartrei belongs to it. However, Ceterach and Phyllitis are, with certainty, the parents of a very interesting fern endemic to the Quarner islands (Yugoslavia) formerly known as Phyllitis hybrida or Scolopendrium hybridum. It is the sole member of the allotetraploid genus Phyllitopsis, described recently by Reichstein. It has been synthesized under experimental conditions and is originated by hybridization between the diploids Ceterach officinarum subsp. bivalens and Phyllitis sagittata followed by chromosome doubling (Fig. 9). The formation of this genus is particularly interesting since it allows us to presume that a process of this kind might have occurred in other families and thus explaining the origin of some genera in which the base number is tetraploid or even hexaploid or octoploid, in comparison with the diploid base number of most genera of the same family.