The linear sequence of genomes exists within the three-dimensional space of the cell nucleus. The spatial arrangement of genes and chromosomes within the interphase nucleus is nonrandom and gives rise to specific patterns. While recent work has begun to describe some of the positioning patterns of chromosomes and gene loci, the structural constraints that are responsible for nonrandom positioning and the relevance of spatial genome organization for genome expression are unclear. Here we discuss potential functional consequences of spatial genome organization and we speculate on the possible molecular mechanisms of how genomes are organized within the space of the mammalian cell nucleus.