We study the optimal power and rate adaptation for cognitive radio (CR) transmission, which maximizes the average spectral efficiency of the secondary user transmitter (ST) subject to average BER constraint for secondary user receiver (SR) and average BER or interference constraint at primary user receivers (PR). Adaptive power and rate adaptation is adopted by the ST to guarantee the performances of both PR and SR, while constant power and fixed modulation is used by the PT. It is shown that in addition to the channel gains of the secondary user (SU), the ST also needs to know the primary user (PU) channel gains, the BER and SINR at the PR. Numerical results on the spectral efficiency of the SU are presented as function of various PU and SU channel gains.Index Terms-cognitive radio (CR) transmission, power and rate adaptation, adaptive modulation, average spectral efficiency, bit error rate (BER).