Cells use actin-based cytoskeletal arrays to accomplish a variety of tasks, including cell division, cell migration, intracellular trafficking, and multicellular tissue morphogenesis. The architectures and assembly dynamics of different actin arrays are closely associated with their different functions. Thus, a key challenge in cell biology is to understand how cells assemble actin arrays with specific architectures at particular places and times. Through decades of effort, cell biologists have identified and characterized the structural components of actin arrays, accessory factors that govern filament assembly, disassembly, crosslink dynamics and motor activity, and upstream signaling pathways that cells use to locally recruit and activate subsets of network elements to initiate the assembly of specific actin arrays. However, the mechanisms by which these components self-organize into specific arrays remain poorly understood. In this thesis, I study the mechanisms that govern self-organization of the contractile ring in the early C. elegans embryo. During cytokinesis, signals from the mitotic apparatus trigger the local assembly of actin filaments and myosin motors at the cell equator, and these rapidly self-organize into a circumferentially aligned array of actin filaments called the contractile ring that constricts to divide the cell. Previous studies showed that in theory, the reorientation of actin filaments by equatorial contraction could explain the rapid emergence of circumferential alignment. Combining single molecule analysis and modeling, I have shown that equatorial filaments turnover far too fast for equatorial contractions to build the observed alignment, even if favorably oriented filaments are selectively stabilized. By tracking the movements of single formin/CYK-1::GFP speckles to monitor the orientation of filament growth in relation to existing actin filaments, I showed that the orientation of equatorial filament growth is biased to favor circumferential alignment. Using multi-color imaging of formin/CYK-1 and a marker for actin filaments, I identified the mechanism for this bias, which I call filament-guided filament assembly (FGFA), in which existing filaments serve as templates to orient the growth of new filaments. Combining modeling and quantitative analysis of CYK-1 trajectories, I showed that FGFA increases the effective lifetime of filament orientation, providing a structural memory of filament orientation that allows slow equatorial contraction to build and maintain highly aligned filament arrays, despite rapid turnover of individual filaments. Finally, I considered one possible mechanism for FGFA, in which dynamic crosslinker PLST-1 rapidly zipper elongating filaments onto existing filaments, allowing them to inherit the same orientation. Combining \textit{in vitro} reconstitution experiments and live imaging of embryos expressing endogenously labeled PLST-1, I showed that PLST-1 is capable of driving FGFA in vitro, and PLST-1 can decorate growing actin filaments fast enough to drive FGFA in vivo. Together, these findings reveal a novel mechanism by which a network of filaments preserves structural information (filament orientation) in the face of rapid turnover of its individual components. This mechanism may underlie the assembly and maintenance of the many other arrays of aligned actin filaments that operate in animal cells.