Adaptation is central to Darwinian evolution, and it may be a key to the survival of species under the conditions of human-induced global change. Despite its importance, many basic questions about the genetic basis of adaptation are still unresolved. This is particularly true for adaptation to gradual change, such as the rise of global temperature and atmospheric CO2, or increase in UV radiation and the concentration of pollutants. To understand the consequences of gradual change, it is important to develop a theory with increased ecological realism. The fundamental event during adaptation is the substitution of a resident allel (i.e., gene variant) by a beneficial mutation. An important goal of current research - both empirical and theoretical - is to learn more about the statistical properties of these substitutions. In particular, much effort is being devoted to understanding the distribution of the effects of new mutations and the distribution of the subset of those mutations that go to fixation and contribute to adaptation. Knowledge of these distributions is necessary to answer seemingly simple questions, such as how many substitutions occur during a typical bout of adaptation - a few with large effects, many with small effects, or a combination of both - and whether they do so in a particular order (e.g., large ones first). One way of addressing these questions in the context of gradual environmental change is to model adaptation with the so-called moving-optimum model. For a single evolving trait, Kopp and Hermisson showed that selection for a moving optimum produces patterns that are fundamentally different from those predicted under constant selection (i.e., after a single, abrupt change in the environment). In the first part of my PhD-project, I extended their model to include multiple characters. In other words, I studied a moving-optimum of Fisher´s classical geometric model of adaptation in high-dimensional trait spaces. In constrast to existing models, this model deals with more complex, yet realistic, biological assumptions on the underlying genetic architecture, in particular genetic correlations between traits. This means that with two or more traits, these can be correlated with respect to selection, mutation or even both, allowing for a variety of different evolutionary outcomes. While it has been shown that more complex organisms pay a "cost of complexity" causing them to adapt more slowly to a single abrupt change in the environment, the generalization of the moving-optimum model to multiple traits likewise enables us to address how organismic complexity (and thereby pleiotropy) influences the populations´ ability to adapt to a sustained environmental change. To analyze this model, we derived analytical approximations for the adaptive process and verified them by means of computer simulations. In particular, we characterized how the fitness of mutations changes over time and obtained approximations for the statistical properties of "adaptive walks" (e.g., the average time and size of adaptive "steps"). Furthermore, we generalized results from previous studies demonstrating that the influence of various genetic and environmental factors on the properties of adaptive walks can be summarized in a single parameter that describes the degree to which adaptation is either "genetically" or "environmentally-limited". Addressing the question how the ability to adapt to changing environment depends on "organismic complexity", we obtained the unexpected result that, even though complexity makes adaptation more difficult, adaptation of complex organisms proceeds in large steps. In addition, we found that patterns of adaptation depend on correlation between traits, and found that the source of the correlations (mutation or selection) has a major effect and shapes the distribution of adaptive substitutions. The exact shape of this distribution, however, strongly depends on the speed of environmental change.