The comparative osteology and phylogenetic relationships of fossil and living paddlefishes (Polyodontidae) are investigated in detail for the first time. This peculiar, poorly known group is of great significance to phylogenetic studies of primitive actinopterygians, because it is one of only two chondrostean families that have survived to the present. Consequently the group is frequently used as an outgroup for studies of the many fossil chondrostean groups and for higher-level studies of actinopterygian (and even osteichthyan) interrelationships.The family has a long but spotty fossil record dating back to the Upper Cretaceous, including species represented by relatively complete skeletons or at least skulls. These relatively complete fossils are extremely rare and belong to the species tPaleopsephurus wilsoni MacAlpin, 1947 from the Upper Cretaceous Hell Creek Formation of Montana (redescribed here); tPolyodon sp. nov. from the lower Paleocene Tullock Formation of Montana (described here); and tCrossopholis magnicaudatus Cope, 1883 from lower Eocene deposits of the Green River Formation in Wyoming (described in detail here for the first time based on much new material). Among the fossils, tCrossopholis is of particular interest, because it is now known by several nearly complete skeletons (reported and described here for the first time). tPaleopsephurus and tPolyodon sp. nov. are known from a single partial skull each, and some fragmentary postcranial material, but are well enough represented to include in our phylogenetic analysis. Other, more fragmentary, Upper Cretaceous and Paleocene material (some newly reported here) is placed in Polyodontidae incertae sedis.There are two extant species in the family, frequently referred to in the literature as "living fossils." These two species are Polyodon spathula from the Mississippi River drainage of North America, and Psephurus gladius primarily from the upper Yangtze River drainage in China. The osteology of Psephurus gladius is described here in detail for the first time.Previous studies of Polyodontidae have been almost exclusively based on Polyodon spathula, because specimens of the other species are extremely rare and very poorly known. After accumulating and preparing material for several years, we were able to describe more fully all polyodontid species and examine the family phylogenetically. Our study reveals that Polyodon spathula is a highly derived member of the family, and generally not a good representative of the family when the family is used as an outgroup for higher-level phylogenetic studies of actinopterygians. The use of Polyodon spathula as the primitive polyodontid has previously led to several serious errors in the literature. We therefore also summarize certain primitive characters of Polyodontidae in this paper.Contrary to previous studies, we show that tPaleopsephurus has the stellate rostral bones unique to polyodontids (in addition to the long median rostral bones and extremely elongate snout also unique to the group). Also contrary to...