The spatial organization of metazoan genomes has a direct influence on fundamental nuclear processes that include transcription, replication, and DNA repair. It is imperative to understand the mechanisms that shape the 3D organization of the eukaryotic genomes. Chromatin insulators have emerged as one of the central components of the genome organization tool-kit across species. Recent advancements in chromatin conformation capture technologies have provided important insights into the architectural role of insulators in genomic structuring. Insulators are involved in 3D genome organization at multiple spatial scales and are important for dynamic reorganization of chromatin structure during reprogramming and differentiation. In this review, we will discuss the classical view and our renewed understanding of insulators as global genome organizers. We will also discuss the plasticity of chromatin structure and its re-organization during pluripotency and differentiation and in situations of cellular stress.