This paper analyses the performance of an underlay cognitive radio network (CRN) over alpha-mu fading channels in terms of symbol error probability (SEP) and effective capacity (EC). The alpha-mu distribution allows for a more general fading model with an increased level of generality compared to well-known models such as Nakagami-m and Rayleigh fading. The SEP and EC at the secondary user receiver are therefore evaluated for different fading parameters alpha and mu In the considered underlay CRN, the secondary user transmit power is subject to a peak interference power constraint in order to limit the interference to the primary user receiver. Given this setting, analytical expressions for the SEP and EC are derived for the general case of alpha-mu fading channels. Numerical results are provided to illustrate the performance of the considered underlay CRN on different fading channels.