Efficiently and accurately simulating partial differential equations (PDEs) in and around arbitrarily defined geometries, especially with high levels of adaptivity, has significant implications for different application domains. A key bottleneck in the above process is the fast construction of a good adaptively-refined mesh. In this work, we present an efficient octree-based adaptive discretization approach capable of carving out arbitrarily shaped regions from the parent domain: an essential requirement for fluid simulations around complex objects and for modeling multi-phase flows. Both explicit and implicit definitions of geometries are supported. We evaluate our approach using a range of applications with varying geometric and physics complexity. These include Large Eddy Simulations (LES) of flows around complex geometries: accurately computing the drag coefficient of a sphere across Reynolds numbers $1− 10^6$ encompassing the drag crisis regime; simulating flow features across a semi-truck for investigating the effect of platooning on efficiency; and a case with multiple complex objects, to evaluate COVID-19 transmission risk in classrooms. We also study problems with more complex physics, solving thermodynamically-consistent Cahn-Hilliard Navier-Stokes system that models two-phase flows with detailed comparisons with results from the literature for canonical cases, including the single bubble rise and Rayleigh-Taylor instability and primary jet atomization.