Quantum systems such as atoms can be used to store information. For example, we can store binary information in two energy levels of an atom by labeling the state with lower energy a "0" and the state with higher energy a "1." However, quantum systems can also exist in superposition states, thereby storing both states of the bit simultaneously, a situation that makes no sense in our ordinary-day experience. This property of quantum bits or "qubits" potentially leads to an exponential increase in memory and processing capacity. It would enable a quantum computer to efficiently solve certain problems such as factorizing large numbers - an ability that could compromise the security of nearly all encryption systems. A quantum computer would also realize an analog of "Schrödinger's Cat," a bizarre situation where a cat could be simultaneously dead and alive. Experiments whose goal is to realize a quantum information processor based on laser manipulations of atoms will be described.