We develop and analyze a set of models for the interaction of HIV with CD4+ T cells. We consider three major subpopulations of T cells: virgin, activated and memory. In our first model we assume that HIV can infect activated cells but not resting cells. We then generalize the model to take into account recent reports that HIV can enter resting cells but that such entry does not lead to the production of completely reverse transcribed copies of the viral genome or integration of the DNA copy into the host cell's genome unless cell activation occurs. Our models show that T-cell memory is greatly reduced by HIV infection and that T-cell depletion may be due to the direct killing of peripheral T cells and T-cell precursors in the thymus.