Colloidal particles with directional interactions that self-assemble into pre-defined structures have the potential to serve as the foundation for a new generation of micro- and nano-structures of remarkable complexity and precision. Dipolar colloid particles tend to align end-to-end and self-assemble into variety of micro- and nano-structures ranging from co-crystals of novel symmetry, to open networks (gels) of cross-linked chains of particles. Quadrupolar colloid particles also tend to self-assemble in a wide variety of structural motifs including sheets, tubes and shells depending upon external conditions. We use molecular dynamics computer simulation to explore the self-assembly, structure, crystallization and/or gelation of systems of colloid particles with permanent dipole moments or quadrupole moments immersed in a high-dielectric solvent. Particle-particle interactions are modeled with discontinuous potentials in order to take advantage of discontinuous molecular dynamics (DMD), a fast simulation technique that is very computationally efficient. We first calculate the phase diagram in the temperature-packing fraction plane of a monodisperse system of dipolar colloid particles using DMD. Several types of phases are found in our simulations: ordered phases including face centered cubic (FCC), hexagonal close packed (HCP) and body centered tetragonal (BCT) at high packing fractions, and fluid, string-fluid and gel phases at low packing fractions. The very low volume fraction gel phases and the well ordered crystal phases are promising for advanced materials applications. We then extend our analysis to a binary mixture of dipolar colloid particles. Phase diagrams for an equimolar binary mixture of dipolar colloid particles with different diameter ratios and different dipole moment ratios are calculated in the temperature-volume fraction plane. These systems exhibit six distinct phases: isotropic fluid, string-fluid, gel, FCC, HCP, BCT, and ten coexisting phases: Fluida + String-fluidb, Fluida + Gelb, String-fluida + Gelb, Gela + BCTb, FCCa + FCCb, FCCa + HCP b, FCCa + FCCb+ Fluid, HCPa + HCP b, BCTa + BCTb, BCTa + BCTb + large voids, depending upon size ratio and dipole moment ratio. An interesting aspect of these phase diagrams is the appearance of co-crystals containing large and small dipolar colloid particles. Even more interesting is the appearance of two unique bicontinuous gel structures - the first gel consists of two independent but interpenetrating networks of cross-linked chains formed by particles with high dipole moment and chains formed by particles with low dipole moment. The second type of gel consists of a network of cross-linked chains formed by particles with high dipole moment; particles with low dipole moment form a sheath around the chains. Such bicontinuous gels may have unusual rheological and transport properties such as multiple yield stresses and multiple percolation temperatures and could form the basis of a new class of soft-solid materials with unique properties and multiple applications. We also explore the structure formation of systems of colloid particles with permanent quadrupole moment. We introduce a simple quadrupole-quadrupole discontinuous potential model that gives rise to the self-organization of surface and tubular structures. We find that the discrete rotational symmetry of a quadrupolar particles gives rise to extended two-dimensional sheet or surface structures that preserve the local symmetry within the organized structure. A new type of anisotropic colloid particle is introduced having a displaced quadrupole moment with a unique symmetry; systems comprised of these particles...