Eukaryotic genomes are packaged into an extensively folded state known as chromatin. Analysis of the structure of eukaryotic chromosomes has been revolutionized by development of a suite of genome-wide measurement technologies, collectively termed "epigenomics." We review major advances in epigenomic analysis of eukaryotic genomes, covering aspects of genome folding at scales ranging from whole chromosome folding down to nucleotide-resolution assays that provide structural insights into protein-DNA interactions. We then briefly outline several challenges remaining and highlight new developments such as single-cell epigenomic assays that will help provide us with a high-resolution structural understanding of eukaryotic genomes.