In this paper, cognitive transmission under quality of service (QoS) constraints is studied. In the cognitive radio channel model, it is assumed that the secondary transmitter sends the data at two different average power levels, depending on the activity of the primary users, which is determined by channel sensing performed by the secondary users. A state-transition model is constructed for this cognitive transmission channel. Statistical limitations on the buffer lengths are imposed to take into account the QoS constraints. The maximum throughput under these statistical QoS constraints is identified by finding the effective capacity of the cognitive radio channel. This analysis is conducted for fixed-power/fixed-rate, fixed-power/variable-rate, and variable-power/variable-rate transmission schemes under different assumptions on the availability of channel side information (CSI) at the transmitter. The impact upon the effective capacity of several system parameters, including channel sensing duration, detection threshold, detection and false alarm probabilities, QoS parameters, and transmission rates, is investigated. The performances of fixed-rate and variable-rate transmission methods are compared in the presence of QoS limitations. It is shown that variable schemes outperform fixed-rate transmission techniques if the detection probabilities are high. Performance gains through adapting the power and rate are quantified and it is shown that these gains diminish as the QoS limitations become more stringent.