This thesis presents a study on the waveform diversity and optimal waveform design approaches for high frequency surface wave (HFSW) radar and multiple-input multiple-output (MIMO) radar. Main contributions of this study are as follows. First, methods for designing single/multiple sparse frequency waveforms with sidelobe constraint are presented. The proposed methods can design digital waveforms that satisfy simultaneously the requirement on Power Spectral Density (PSD) and the requirement on sidelobe in correlation function, i.e., autocorrelation function for single waveform design and both autocorrelation functions and cross-correlation functions for multiple waveforms design. Objective functions are constructed based on the PSD and sidelobe requirements and are minimized by two optimization approaches. These two approaches together can provide much flexibility in designing sparse frequency waveforms for different applications. Some implementation issues such as quantization effect, property of Low Probability of Intercept, and Doppler Tolerance are discussed as well. Second, some basic issues regarding to space-time adaptive processing (STAP) in MIMO radar with waveform diversity are investigated in order to predict the performance of MIMO STAP.