The dynamics of adaptation are difficult to predict because it is highly stochastic even in large populations. The uncertainty emerges from random genetic drift arising in a vanguard of particularly fit individuals of the population. Several approaches have been developed to analyze the crucial role of genetic drift on the expected dynamics of adaptation, including the mean fitness of the entire population, or the fate of newly arising beneficial deleterious mutations. However, little is known about how genetic drift causes fluctuations to emerge on the population level, where it becomes palpable as variations in the adaptation speed and the fitness distribution. Yet these phenomena control the decay of genetic diversity and variability in evolution experiments and are key to a truly predictive understanding of evolutionary processes. Here, we show that correlations induced by these emergent fluctuations can be computed at any arbitrary order by a suitable choice of a dynamical constraint. The resulting linear equations exhibit fluctuationinduced terms that amplify short-distance correlations and suppress long-distance ones. These terms, which are in general not small, control the decay of genetic diversity and, for wave-tip dominated ("pulled") waves, lead to anticorrelations between the tip of the wave and the lagging bulk of the population. While it is natural to consider the process of adaptation as a branching random walk in fitness space subject to a constraint (due to finite resources), we show that other traveling wave phenomena in ecology and evolution likewise fall into this class of constrained branching random walks. Our methods, therefore, provide a systematic approach toward analyzing fluctuations in a wide range of population biological processes, such as adaptation, genetic meltdown, species invasions, or epidemics.KEYWORDS traveling-wave models; asexual adaptation; Fisher waves; tuned models; stochastic modeling; higher-order correlation functions; exact approaches M ANY important evolutionary and ecological processes rely on the behavior of a small number of individuals that have a large dynamical influence on the population as a whole. This is, perhaps, most obvious in the case of biological adaptation in asexual species, such as microbes (over timescales where horizontal gene transfer is irrelevant). Future generations descend from a small number of currently welladapted individuals. The genetic footprint of the large majority of the population is wiped out over time by the fixation of more fit genotypes. For a large influx of beneficial mutations, these dynamics can be visualized as a traveling wave in fitness space; see Figure 1A. At any time, the currently most fit "pioneer" individuals reside in the small tip of the wave. As time elapses, the wave moves toward higher fitness and the formerly rare most fit individuals dominate the population. By that time, however, a new wave tip of even more fit mutants has formed by mutations and the cycle of transient dominance continues. Evolu...