Though there is an increasing support for an important contribution of copy number variation (CNV) to the genetic architecture of complex disease, few methods have been developed for the analysis of such variation in the context of genetic association studies. In this paper, we propose a generalization of family-based association tests (FBATs) to allow for the analysis of CNVs at a genome-wide level. We translate the popular FBAT approach so that, instead of genotypes, raw intensity values that reflect copy number are used directly in the test statistic, thereby bypassing the need for a CNV genotyping algorithm. Moreover, both inherited and de novo CNVs can be analyzed without any prior knowledge about the type of CNV, making it easily applicable to large-scale association studies. All robustness properties of the genotype FBAT approach are maintained and all previously developed FBAT extensions, including FBATs for time-to-onset, multivariate FBATs, and FBAT-testing strategies, can be directly transferred to the analysis of CNVs. Using simulation studies, we evaluate the power and the robustness of the new approach. Furthermore, for those CNVs that can be genotyped, we compare FBATs based on genotype calls with FBATs that are directly based on the intensity data. An application to one of the first CNV genome-wide-association studies of asthma identifies a very plausible candidate gene. A software implementation of the approach is freely available at http://www.hsph.harvard.edu/research/iuliana-ionita/software. The approach has also been completely integrated in the PBAT software package.