This thesis discusses the design, fabrication, and characterization of photonic crystal cavities with embedded InGaAs quantum dots. Cavities with different geometries are investigated, including H1, H2, L3 and L5. The main focus is on H2 type, consisting of a defect formed by omitting seven air holes in the center of a triangular lattice. The design and simulation of the cavities are performed by using the Finite-Difference Time-Domain method. It is found that by engineering the air holes surrounding the cavity, the quality factor can be increased significantly by the gentle mode confinement method. The quantum dot samples are grown by using molecular beam epitaxy technique. Then, the photonic crystals are produced by using electron beam lithography and etching techniques. The fabrication process is developed and optimized in order to obtain high quality GaAs photonic crystal membranes. The cavities are characterized by using photoluminescence technique at low temperature. Polarization-dependent measurements are also performed in order to identify the cavity modes. The results are in good agreement with our theoretical calculations. Finally, the p-shell Rabi oscillations of a single quantum dot in a modified H2 photonic crystal cavity are investigated.