Three-dimensional genome structure plays an important role in gene regulation. Globally, chromosomes are organized into active and inactive compartments while, at the gene level, looping interactions connect promoters to regulatory elements. Topologically associating domains (TADs), typically several hundred kilobases in size, form an intermediate level of organization. Major questions include how TADs are formed and how they are related to looping interactions between genes and regulatory elements. Here we performed a focused 5C analysis of a 2.8 Mb chromosome 7 region surrounding CFTR in a panel of cell types. We find that the same TAD boundaries are present in all cell types, indicating that TADs represent a universal chromosome architecture. Furthermore, we find that these TAD boundaries are present irrespective of the expression and looping of genes located between them. In contrast, looping interactions between promoters and regulatory elements are cell-type specific and occur mostly within TADs. This is exemplified by the CFTR promoter that in different cell types interacts with distinct sets of distal cell-type-specific regulatory elements that are all located within the same TAD. Finally, we find that long-range associations between loci located in different TADs are also detected, but these display much lower interaction frequencies than looping interactions within TADs. Interestingly, interactions between TADs are also highly cell-type-specific and often involve loci clustered around TAD boundaries. These data point to key roles of invariant TAD boundaries in constraining as well as mediating cell-type-specific long-range interactions and gene regulation.