The cytoskeleton is a collection of protein assemblies that dynamically impose spatial structure in cells and coordinate processes such as cell division and mechanical regulation. Biopolymer filaments, cross-linking proteins, and enzymatically active motor proteins collectively self-organize into various precise cytoskeletal assemblies critical for specific biological functions. An outstanding question is how the precise spatial organization arises from the component macromolecules. In this talk, I will focus on a new system to investigate simple physical mechanisms of self-organization in biological assemblies. Using a minimal set of purified proteins, we create assemblies of cross-linked biopolymer filaments that we find are liquid droplets rather than the expected entangled aggregates. Microscopy experiments, together with a continuum model based on liquid crystal physics, reveal that these droplets are anisotropic liquids, where cross-linking modulates droplet shape and liquid properties. Through the addition of enzymatically active motor proteins we construct composite assemblies, evocative of cellular structures such as spindles and sarcomeres, where the inherent anisotropy drives motor self-organization and droplet deformation. Our results have the potential to broadly expand our understanding of the physical principles guiding self-organization in other biological assemblies and inform bio-inspired materials design.