The nucleus is the control center of the cell, containing genetic material essential for life. Yet, the physical principles underlying its organization in space and time remain a mystery. In this talk, I will show work from my group revealing that the biophysical complexity of the nucleus can be organized around three inter-related and interactive facets: heterogeneity, activity and rheology. Most nuclear constituents are sites of active, ATP-dependent processes and are thus inherently dynamic: The genome undergoes constant rearrangement, the nuclear envelope flickers and fluctuates, nucleoli migrate and coalesce, and many of these events are mediated by nucleoplasmic flows and interactions. And yet there is spatiotemporal organization in terms of hierarchical structure of the genome, its coherently-moving regions and membrane-less compartmentalization via phase-separated nucleoplasmic constituents. Moreover, the non-equilibrium or activity-driven nature of the nucleus gives rise to emergent rheology and material properties that impact all cellular processes via the central dogma of molecular biology. New biophysical insights into the cell nucleus can come from appreciating this rich inner life.