Robust cut-cell algorithms are detailed for general multi-level Cartesian grid direct simulation Monte Carlo (DSMC) implementations. The essential components of the cut-cell algorithms, necessary to achieve the overall functionality of general multi-level Cartesian grid DSMC implementation, are discussed in detail. Different approaches for each component of the cut-cell algorithms are elaborated and compared, based on time effciency, robustness and ease of implementation. Special situations that arise in sorting the surface triangles, calculating the cut-cell volume, and identifying multiple cut-cells resulting from single regular Cartesian cell, are handled in the algorithms, and their effects on the overall simulation results are analyzed using test cases. Simulations for a Mach number 20.2 N2 gas flowing over a spherical-blunted Planetary Probe are conducted as one demonstration, and overall good agreements with experimental results are achieved for heat flux over the Planetary Probe surface. Rarefied flow over a geometry resembling the MIR Space Station is also conducted, to demonstrate the ability of the proposed cut-cell algorithms in handling complex geometries. The time required to cut the background Cartesian cells from the surface mesh and to calculate the volume of the resulting cut-cells is shown to be comparable to the time cost for one DSMC simulation time step for cases considered.