Actin and microtubule filaments, with their auxiliary proteins, enable the cytoskeleton to perform vital processes in the cell by tuning the organizational, mechanical properties and dynamics of the network. Despite their critical importance and interactions in cells, we are only beginning to uncover information about the composite network. Here, I use florescence microscopy to explore the role of filaments characteristics, interactions and activities in the self-organization and steady-state dynamics of the composite network of filaments. First, I discuss active self-organization of semiflexible actin and rigid microtubule filaments in the 2D composite network while myosin II and kinesin-1 motor proteins propel actin and microtubule filaments, respectively. Second, I studied the steady-state mobility of the 3D composite network is studied when the interactions of filaments are regulated by the varying amount of crosslinkers. In a composite network where only actin filaments crosslinked using biotin-NeutrAvidin molecules, microtubule mobility is tuned by actin crosslinking and displays non-monotonic dependence on the amount of actin crosslinkers. Third, I included antiparallel microtubule crosslinkers, MAP65, as well as biotin-NeutrAvidin actin crosslinkers to reveal the different roles of these crosslinkers in the structure and mobility of the composite network. While actin crosslinkers dictated the mobility, microtubule crosslinkers control the co-localization of filaments. Finally, I worked on an active composite network of actin, microtubule, and myosin II motor proteins. The structural changes in the contractile composite network is characterized using correlation length measurements. These results provide a valuable insight into the cytoskeletal filaments interactions and their vital roles in various biological processes in cells. Furthermore, this knowledge could enable us to design autonomous bioinspired materials with tunable mechanical properties.