An increasing number of studies demonstrate phenotypic and genetic changes in natural populations that are subject to climate change, and there is hope that some of these changes will contribute to avoiding species extinctions (‘evolutionary rescue’). Here, we review theoretical models of rapid evolution in quantitative traits that can shed light on the potential for adaptation to a changing climate. Our focus is on quantitative‐genetic models with selection for a moving phenotypic optimum. We point out that there is no one‐to‐one relationship between the rate of adaptation and population survival, because the former depends on relative fitness and the latter on absolute fitness. Nevertheless, previous estimates that sustainable rates of genetically based change usually do not exceed 0.1 haldanes (i.e., phenotypic standard deviations per generation) are probably correct. Survival can be greatly facilitated by phenotypic plasticity, and heritable variation in plasticity can further speed up genetic evolution. Multivariate selection and genetic correlations are frequently assumed to constrain adaptation, but this is not necessarily the case and depends on the geometric relationship between the fitness landscape and the structure of genetic variation. Similar conclusions hold for adaptation to shifting spatial gradients. Recent models of adaptation in multispecies communities indicate that the potential for rapid evolution is strongly influenced by interspecific competition.

The large body of theory on speciation with gene flow has brought to light fundamental differences in the effects of two types of mating rules on speciation: preference/trait rules, in which divergence in both (female) preferences and (male) mating traits is necessary for assortment, and matching rules, in which individuals mate with like individuals on the basis of the presence of traits or alleles that they have in common. These rules can emerge from a variety of behavioral or other mechanisms in ways that are not always obvious. We discuss the theoretical properties of both types of rules and explain why speciation is generally thought to be more likely under matching rather than preference/trait rules. We furthermore discuss whether specific assortative mating mechanisms fall under a preference/trait or matching rule, present empirical evidence for these mechanisms, and propose empirical tests that could distinguish between them. The synthesis of the theoretical literature on these assortative mating rules with empirical studies of the mechanisms by which they act can provide important insights into the occurrence of speciation with gene flow. Finally, by providing a clear framework we hope to inspire greater alignment in the ways that both theoreticians and empiricists study mating rules and how these rules affect speciation through maintaining or eroding barriers to gene flow among closely related species or populations.

Adaptation lies at the heart of Darwinian evolution. Accordingly, numerous studies have tried to provide a formal framework for the description of the adaptive process. Out of these, two complementary modelling approaches have emerged: While so-called adaptive-walk models consider adaptation from the successive fixation of de-novo mutations only, quantitative genetic models assume that adaptation proceeds exclusively from pre-existing standing genetic variation. The latter approach, however, has focused on short-term evolution of population means and variances rather than on the statistical properties of adaptive substitutions. Our aim is to combine these two approaches by describing the ecological and genetic factors that determine the genetic basis of adaptation from standing genetic variation in terms of the effect-size distribution of individual alleles. Specifically, we consider the evolution of a quantitative trait to a gradually changing environment. By means of analytical approximations, we derive the distribution of adaptive substitutions from standing genetic variation, that is, the distribution of the phenotypic effects of those alleles from the standing variation that become fixed during adaptation. Our results are checked against individual-based simulations. We find that, compared to adaptation from de-novo mutations, (i) adaptation from standing variation proceeds by the fixation of more alleles of small effect; (ii) populations that adapt from standing genetic variation can traverse larger distances in phenotype space and, thus, have a higher potential for adaptation if the rate of environmental change is fast rather than slow.