Over two thousand plant species have been modified morphologically through cultivation and human use. Here, we review three aspects of crop domestication that are currently undergoing marked revisions, due to analytical advancements and their application to whole genome resequencing (WGS) data. We begin by discussing the duration and demographic history of domestication. There has been debate as to whether domestication occurred quickly or slowly. The latter is tentatively supported both by fossil data and application of WGS data to sequentially Markovian coalescent methods that infer the history of effective population size. This history suggests the possibility of extended human impacts on domesticated lineages prior to their purposeful cultivation. We also make the point that demographic history matters, because it shapes patterns and levels of extant genetic diversity. We illustrate this point by discussing the evolutionary processes that contribute to the empirical observation that most crops examined to date have more putatively deleterious alleles than their wild relatives. These deleterious alleles may contribute to genetic load within crops and may be fitting targets for crop improvement. Finally, the same demographic factors are likely to shape the spectrum of structural variants (SVs) within crops. SVs are known to underlie many of the phenotypic changes associated with domestication and crop improvement, but we currently lack sufficient knowledge about the mechanisms that create SVs, their rates of origin, their population frequencies and their phenotypic effects.