The comparative osteology, phylogenetic relationships, and historical biogeography of all known taxa of fossil and living amiid fishes (Halecomorphi: Amiidae) are investigated in detail. Previously, the detailed osteology of nearly all fossil amiids was unknown. We present the first well-supported comprehensive phylogeny for fossil and living amiid fishes. We synthesize clearly documented phylogenetic data on amiids and other halecomorph fishes with other historical phenomena such as ontogeny, historical biogeography, stratigraphic paleontology, and paleoecology (both “stationary” and “historical”). We also use our study of halecomorph fishes as a platform to explore several fundamental methodological and theoretical concepts important to phylogenetic/evolutionary investigations. These concepts pertain mainly to (1) the use of comparative empirical data to interpret various historical patterns and (2) the practice of integrating fossil and living species together in original (i.e., non-literature-based) phylogenetic studies.As proposed by Patterson (1973), Amia calva is the sole surviving member of a formerly diverse clade, Halecomorphi. Even Amiidae (the halecomorph subgroup that is the focus of this study) was taxonomically diverse and widespread during Mesozoic and Paleogene times. Amiidae here includes only those amiiform fishes that have solid perichordally ossified diplospondylous centra of both the “normal” (a term used here for non-alternating) and the alternating type in the caudal region. Consequently, †Sinamiidae and †Caturoidea (†Caturidae plus †Liodesmidae) are excluded from the family Amiidae. To set the style and terminology for description of the fossil taxa, and to present a fresh look at the osteology and development of the only living amiiform, we redescribe Amia calva in detail using modern methods of preparation and illustration. Detailed study of Amia calva is critical for interpreting probable intraspecific variation and ontogeny in fossil skeletons. We examined individual variation of both bone morphology and developmental timing in Amia calva.We classify the family Amiidae into four subfamilies here: †Amiinae (including †Amia, †Cyclurus, †Pseudamiatus, and † “Amia” hesperia), †Vidalamiinae subfam. nov. (including † Vidalamia, †Pachyamia, †Melvius, †Calamopleurus, and †Maliamia), †Solnhofenamiinae subfam. nov. (including †Solnhofenamia gen. nov.), and †Amiopsinae subfam. nov. (including †Amiopsis). The genus †Nipponamia remains Amiidae subfamily indeterminate until additional material can be found and studied, although it does not appear to belong to Amiinae or †Amiopsinae subfam. nov. as defined here. Other nominal genera of Amiidae are discussed as nomina nuda or nomina dubia.We found Amiinae to be a clearly monophyletic group diagnosed by numerous synapomorphic characters. This is the only amiid (and halecomorph) subfamily known to have survived beyond the early Eoccne. It was a diverse group with a known stratigraphic range of Late Cretaceous through the present. The earliest diagnosable species in the subfamily is †Cyclurus fragosus from the upper Maastrichtian of Alberta, Canada, and the earliest Amiinae indeterminate is a collection of fragments from the upper Cenomanian of Uzbekhistan (“?Amia semimarina” Nessov 1985, a nomen dubium). Earlier forms reported by Boreske (1974) could not be verified as Amiinae. Amiinae was widespread and particularly speciose during the Eocene, and appears to have been confined almost exclusively to fresh water. The extremely rare amiine fossil fragments from brackish water deposits probably represent accidental or, at most, occasional migration into brackish environments or possibly entrapment in a drying water system. Amiinae includes three valid species of Amia (including †A. pattersoni sp. nov.), eight valid species of †Cyclurus (all of which are described here), and † “Amia” hesperia (a valid species of uncertain generic affinity). It also includes the genus †Pseudamiatus which remains †Amiinae indeterminate, until the type and only known specimen can be further prepared or until additional material can be found. It probably belongs in Amia or †Cyclurus as those genera are defined here. We also review more than 30 other nominal species of amiines, all of which we believe are invalid names (either nomina dubia, nominanuda, or subjective junior synonyms) but some of which nevertheless extend the stratigraphic and geographic range of the subfamily Amiinae. Although we clarify the known diversity and monophyly of Amiinae, and add new morphological information about numerous amiine species, further materials and work are needed to resolve phylogenetic interrelationships within this subfamily.†Vidalamiinae subfam. nov. is a previously unrecognized, diverse group of amiid taxa, several of which reach very large sizes (to about two meters total length). This well-supported monophyletic subfamily is known only from the Early Cretaceous to the early Eocene, and contains two well-supported monophyletic subgroups: †Vidalamiini tribe nov. and †Calamopleurini tribe nov. These tribes have interesting biogeographic distributions that seem incongruent with today's geography, but these distributions make intuitive sense given current interpretations of the paleogeography of the Early Cretaceous (illustrated here). †Vidalamiini tribe nov. includes †Vidalamia (one species from coastal estuarine/lagoonal deposits of Spain), †Pachyamia (two species: one from marine deposits of the Middle East and a new species from marine deposits in Mexico) and †Melvius (two species from deposits along the margin of the Upper Cretaceous North American Seaway). During the Early Cretaceous, this seemingly far-flung distribution was connected by a continuous shallow marine continental margin and seaway. All five species of †Vidalamiini tribe nov. are described in detail here. †Calamopleurini tribe nov. includes †Calamopleurus (three species: one from an inconclusively determined paleoenvironment of eastern Brazil, one species from a probable freshwater paleoenvironment of eastern Brazil, and a new species from a yet unknown paleoenvironment of Morocco) and †Maliamia (one species from west-central Africa in deposits of unknown water chemistry). Most of these species (particularly †C. cylindricus) are described in detail here. †Vidalamiinae subfam. nov. is the sister group to Amiinae, and the two subfamilies together form the supersubfamily Amiista nuper interposita. The term “nuper interposita” is introduced here to indicate the first published use of new names inserted between existing family group names.†Solnhofenamiinae subfam. nov. is a Late Jurassic marine group from Europe (Germany and France) currently containing a single species, †Solnhofenamia elongata gen. nov. This species was formerly placed in the genus † Urocles as † “Urocles” elongatus. However, the genus † Urocles is invalid in our scheme because the type species of †Urocles is congeneric with †Amiopsis, and the name †Amiopsis has priority. We find that † “Urocles” elongatus is more closely related to a group containing Amiinae plus †Vidalamiinae subfam. nov., than it is to the type species of the genus †Urocles (= †Amiopsis lepidota here). Therefore, we place †“Urocles” elongatus in the new genus †Solnhofenamia. Because †S. elongata gen. nov. is the sister group to a group containing Amiinae plus †Vidalamiinae subfam. nov., we put it into †Solnhofenamiinae subfam. nov. We describe this species in detail for the first time.†Amiopsinae subfam. nov. is a poorly supported stem group of amiids from Late Jurassic to late Cretaceous (Turonian) marine and freshwater deposits of Europe (Yugoslavia, Germany, Spain, England, Belgium, and possibly France). This group, with five valid species, may be non-monophyletic. †Amiopsinae subfam. nov. contains most of the amiids excluded from the hypersubfamily Amiida nuper interposita, a monophyletic group containing ((Amiinae + †Vidalamiinae subfam. nov.) + †Solnhofenamiinae subfam. nov.). All five species of †Amiopsis are described here with much new detail.The phylogenetic position and biogeographic significance of the single known, poorly preserved specimen of †Nipponamia, from Early Cretaceous freshwater deposits of Japan, is largely unknown. Additional material is needed to better understand its morphology.In addition to our cladogram of Amiidae, we produced a cladogram of Halecomorphi. Our sample of non-amiid halecomorph taxa was far less complete than our sample of amiid taxa; nevertheless we found several well-supported nodes within non-amiid halecomorphs which can serve as general guides for future research.We discuss numerous general methodological, philosophical, and theoretical concepts in this paper, including the construction of taxonomic diagnoses, the treatment of undiagnosable taxa, and the treatment of ontogenetically variable characters. We review the effects on phylogenetic analysis of problematic (e.g., “missing”) data entries resulting from either problematic taxa (e.g., highly incomplete fossil taxa) or problematic characters (characters whose states are unknown for most taxa). Various types of problematic data are also reviewed. We discuss the exclusion of certain problematic taxa (here termed “inserted” taxa) and problematic characters (here termed “mapped” characters) from phylogenetic analyses. We also examine the effects of phylogenetically “redundant” taxa and consider the comparative value of wellresolved nodes versus completely resolved trees in cladistic studies. These concepts and terms can all be located in this volume by using the table of contents or the subject index.Our study exemplifies the large amount of natural historical information that can be interpreted from multidisciplinary pattern studies of fossil and living actinopterygian fishes. We term our general approach to the study of natural history as “an empirical synthetic pattern approach”, an exploratory method whose fundamentals go back to the 19 century naturalist Louis Agassiz (1807–1873). Many other actinopterygian groups of equal potential remain virtually unstudied with regard to their broad historical significance (e.g., phylogenetic pattern congruence between comparative morphology, ontogeny, biogeography, and stratigraphy). Exceptional preservation and abundant poorly known material make fossil actinopterygians one of the greatest remaining frontiers in vertebrate paleontology.