We study the performance of cognitive radio networks (CRNs) when incorporating adaptive modulation at the physical layer. Three types of CRNs are considered, namely opportunistic spectrum access (OSA), spectrum sharing (SS) and sensing-based SS. We obtain closed-form expressions for the average spectral efficiency achieved at the secondary network and the optimal power allocation for both continuous and discrete rate types of adaptive modulation assuming perfect channel state information. The obtained numerical results show the achievable performance gain in terms of average spectral efficiency and the impact on power allocation when adaptive modulation is implemented at the physical layer that is due to the effect of the cut-off level that is determined from the received signal-to-noise ratio for each CRN type. The performance assessment is taking place for different target bit error rate values and fading regions, thereby providing useful performance insights for various possible implementations.