Eukaryotes and archaebacteria form the clade neomura and are sisters, as shown decisively by genes fragmented only in archaebacteria and by many sequence trees. This sisterhood refutes all theories that eukaryotes originated by merging an archaebacterium and an alpha-proteobacterium, which also fail to account for numerous features shared specifically by eukaryotes and actinobacteria. I revise the phagotrophy theory of eukaryote origins by arguing that the essentially autogenous origins of most eukaryotic cell properties (phagotrophy, endomembrane system including peroxisomes, cytoskeleton, nucleus, mitosis and sex) partially overlapped and were synergistic with the symbiogenetic origin of mitochondria from an alpha-proteobacterium. These radical innovations occurred in a derivative of the neomuran common ancestor, which itself had evolved immediately prior to the divergence of eukaryotes and archaebacteria by drastic alterations to its eubacterial ancestor, an actinobacterial posibacterium able to make sterols, by replacing murein peptidoglycan by N-linked glycoproteins and a multitude of other shared neomuran novelties. The conversion of the rigid neomuran wall into a flexible surface coat and the associated origin of phagotrophy were instrumental in the evolution of the endomembrane system, cytoskeleton, nuclear organization and division and sexual life-cycles. Cilia evolved not by symbiogenesis but by autogenous specialization of the cytoskeleton. I argue that the ancestral eukaryote was uniciliate with a single centriole (unikont) and a simple centrosomal cone of microtubules, as in the aerobic amoebozoan zooflagellate Phalansterium. I infer the root of the eukaryote tree at the divergence between opisthokonts (animals, Choanozoa, fungi) with a single posterior cilium and all other eukaryotes, designated 'anterokonts' because of the ancestral presence of an anterior cilium. Anterokonts comprise the Amoebozoa, which may be ancestrally unikont, and a vast ancestrally biciliate clade, named 'bikonts'. The apparently conflicting rRNA and protein trees can be reconciled with each other and this ultrastructural interpretation if long-branch distortions, some mechanistically explicable, are allowed for. Bikonts comprise two groups: corticoflagellates, with a younger anterior cilium, no centrosomal cone and ancestrally a semi-rigid cell cortex with a microtubular band on either side of the posterior mature centriole; and Rhizaria [a new infrakingdom comprising Cercozoa (now including Ascetosporea classis nov.), Retaria phylum nov., Heliozoa and Apusozoa phylum nov.], having a centrosomal cone or radiating microtubules and two microtubular roots and a soft surface, frequently with reticulopodia. Corticoflagellates comprise photokaryotes (Plantae and chromalveolates, both ancestrally with cortical alveoli) and Excavata (a new protozoan infrakingdom comprising Loukozoa, Discicristata and Archezoa, ancestrally with three microtubular roots). All basal eukaryotic radiations were of mitochondrial aerobes; hydrogenosomes evolved polyphyletically from mitochondria long afterwards, the persistence of their double envelope long after their genomes disappeared being a striking instance of membrane heredity. I discuss the relationship between the 13 protozoan phyla recognized here and revise higher protozoan classification by updating as subkingdoms Lankester's 1878 division of Protozoa into Corticata (Excavata, Alveolata; with prominent cortical microtubules and ancestrally localized cytostome--the Parabasalia probably secondarily internalized the cytoskeleton) and Gymnomyxa [infrakingdoms Sarcomastigota (Choanozoa, Amoebozoa) and Rhizaria; both ancestrally with a non-cortical cytoskeleton of radiating singlet microtubules and a relatively soft cell surface with diffused feeding]. As the eukaryote root almost certainly lies within Gymnomyxa, probably among the Sarcomastigota, Corticata are derived. Following the single symbiogenetic origin of chloroplasts in a corticoflagellate host with cortical alveoli, this ancestral plant radiated rapidly into glaucophytes, green plants and red algae. Secondary symbiogeneses subsequently transferred plastids laterally into different hosts, making yet more complex cell chimaeras--probably only thrice: from a red alga to the corticoflagellate ancestor of chromalveolates (Chromista plus Alveolata), from green algae to a secondarily uniciliate cercozoan to form chlorarachneans and independently to a biciliate excavate to yield photosynthetic euglenoids. Tertiary symbiogenesis involving eukaryotic algal symbionts replaced peridinin-containing plastids in two or three dinoflagellate lineages, but yielded no major novel groups. The origin and well-resolved primary bifurcation of eukaryotes probably occurred in the Cryogenian Period, about 850 million years ago, much more recently than suggested by unwarranted backward extrapolations of molecular 'clocks' or dubious interpretations as 'eukaryotic' of earlier large microbial fossils or still more ancient steranes. The origin of chloroplasts and the symbiogenetic incorporation of a red alga into a corticoflagellate to create chromalveolates may both have occurred in a big bang after the Varangerian snowball Earth melted about 580 million years ago, thereby stimulating the ensuing Cambrian explosion of animals and protists in the form of simultaneous, poorly resolved opisthokont and anterokont radiations.