Since the late 1990s, the characterization of complete DNA sequences for a large and taxonomically diverse set of species has continued to gain in speed and accuracy. Sequence analyses have indicated a strikingly baroque structure for most eukaryotic genomes, with multiple repeats of DNA sequences and with very little of the DNA specifying proteins. Much of the DNA in these genomes has no known function. These results have generated strong interest in the factors that govern the evolution of genome architecture. (...) While adaptationist ‘just so’ stories have been offered, recent theoretical analyses based on mathematical population genetics strongly suggest that non-adaptive processes dominate genome architecture evolution. This article critically synthesizes and develops these arguments, explicating a core argument along with several variants. It provides a critical assessment of the evidence that supports these arguments’ premises. It also analyses adaptationist responses to these arguments and notes potential problems with the core argument. These theoretical analyses continue the molecular reinterpretation of evolution initiated by the neutral theory in 1968. The article ends by noting that some of these arguments can also be extended to evolution at higher levels of organization which raises questions about adaptationism in general. This remains a puzzle because there is probably little reason to doubt that many organismic features are genuine adaptations. 1 Introduction2 Preliminaries: Senses of Adaptationism3 Genome Architecture3.1 Surprises of early eukaryotic genetics3.2 Genome structure, post-20014 The Case against Adaptationism4.1 Just so stories versus population genetics4.2 The core argument4.3 Three variants of the core argument4.4 Examples: Non-adaptive features of the genome5 Adaptationist Responses6 Concluding Remarks. (shrink)