A particle gel is a network of aggregated colloidal particles with soft solid-like mechanical properties. Its structural and rheological properties, and the kinetics of its formation, are dependent on the sizes and shapes of the constituent particles, the volume fraction of the particles, and the nature of the interactions between the particles before, during and after gelation. Particle gels may be permanent or transient depending on whether the colloidal forces between the aggregating particles lead to irreversible bonding or weak reversible interactions. With short-range reversible interactions, network formation is typically associated with phase separation or kinetic arrest due to particle crowding. Much existing knowledge has been derived from computer simulations of idealized model systems containing spherical particles interacting with well-defined pair potentials. The status of current progress is reviewed here by summarizing the underlying methodology and key findings from a range of simulation approaches: Monte Carlo, molecular dynamics, Brownian dynamics, Stokesian dynamics, dissipative particle dynamics, multiparticle collision dynamics, and fluid particle dynamics. Then it is described how the technique of Brownian dynamics simulation, in particular, has provided detailed insight into how different kinds of bonding and weak reversible interactions can affect the aggregate fractal structure, the percolation behaviour, and the small-deformation rheological properties of network-forming colloidal systems. A significant ongoing development has been the establishment and testing of efficient algorithms that are able to capture the subtle dynamic structuring effects that arise from effects of interparticle hydrodynamic interactions. This has led to an appreciation recently of the potentially important role of these particle-particle hydrodynamic effects in controlling the evolving morphology of simulated colloidal aggregates and in defining the location of the sol-gel phase boundary.