Evolvability is the ability of a system or population to respond to selection by producing heritable and selectable phenotypic variation. In contrast robustness is the ability of a phenotype to persist against perturbations. By definition a system cannot be evolvable and robust at the same time. However, evolvability and robustness are both important properties to evolve complex traits. This creates a paradox for the evolution of complex phenotypes. Evolvability depends on the way how genetic variation translates into phenotypic variation. This process is the key to understand the relationship between evolvability and robustness in complex organisms and how they shape evolutionary change. I am using two different types of mathematical models of the genotype-phenotype map to explore the relationship between evolvability and robustness. I am able to show that the relationship between evolvability and robustness depends on the topology of the genotype-phenotype map using a Boolean genotype-phenotype map. I am investigating more complex genotype-phenotype maps in a population-genetics context using reaction-diffusion models of pattern formation that are motivated by the development of butterfly eyespots. The reaction-diffusion model is used as genetic architecture in individual-based simulations of populations to study the influence of selection strength on evolvability and robustness. I am exploring the ability of the system to produce and maintain genetic variation over long-term evolutionary change and identify processes of pattern formation that facilitate evolvability and can give insight in the origination of novel patterns over time.