Over the last two decades, the system of exciton-polaritons (polaritons) in a semiconductor microcavity has become an important platform for studying the physics of quantum fluids in a solid-state system. Polaritons are formed by the strong coupling between photons and a sharp electronic resonance (e.g. an exciton resonance) in a cavity. They are interacting bosonic particles with a small effective mass due to their half-light and half-matter nature. Spontaneous coherence phenomena, such as the superfluid transition and Bose-Einstein condensation (BEC), have been observed in polariton systems at temperatures in the range from several Kelvin to room temperature. This dissertation focuses on new methods of trapping polaritons and the BEC and superfluidity of polaritons in these new traps. The first part of this dissertation describes experiments on trapping polaritons with an optically generated potential barrier. When the polariton density increases, there is a transition from ballistic motion to coherent motion of polaritons over hundreds of micrometers. At even higher particle density, there is a very sharp transition from the coherent motion state to the ground state of the trap. The second part of this dissertation explores the superfluid properties of polaritons in a ring-shaped trap. This ring trap is formed by combining a stress-induced harmonic trap with an optically created barrier at the trap center. This trapping method enables fine control of the trap profile as well as the properties of the polaritons in the trap. The formation of a polariton ring condensate is observed in this trap. The phase and polarization measurement of the ring condensate reveals that it is in a half-quantized circulation state which features a phase shift of π and a polarization vector rotation of π of the polaritons around a closed path in the ring. The direction of the circulation of the flow around the ring fluctuates randomly between clockwise and counter-clockwise from one shot to the next. In contrast, the rotation of the polarization of polaritons is very stable. This property is experimentally studied, and it is found that the stable spatial polarization pattern may relate to the optical spin Hall effect
