There has been a growing interest in spatial sound generation arising from the development of new communications and media technologies. Binaural spatial sound systems are capable of encoding and rendering sound sources accurately in three dimensional space using only two recording/playback channels. This is based on the concept of the Head-Related Transfer Function (HRTF), which is a set of acoustic filters from the sound source to a listener's eardrums and contains all the listening cues used by the hearing mechanism for decoding spatial information encoded in binaural signals. The HRTF is usually obtained from acoustic measurements on different persons. In the case of discrete data and sets of measurements corresponding to different human subjects, it is desirable to have a continuous functional representation of the HRTF for efficiently rendering moving sounds in the virtual spatial audio systems; further this representation should be well-suited for customization to an individual listener. In this thesis, modal analysis is applied to examine the HRTF data structure, that is to employ the wave equation solutions to expand the HRTF with separable basis functions. This leads to a general representation of the HRTF into separated spatial and spectral components, where the spatial basis functions modes account for the HRTF spatial variations and the remaining HRTF spectral components provide a new means to examine the human body scattering behavior. The general model is further developed into the HRTF continuous functional representations. We use the normalized spatial modes to link near-field and far-field HRTFs directly, which provides a way to obtain the HRTFs at different ranges from measurements conducted at only a single range. The spatially invariant HRTF spectral components are represented continuously using an orthogonal series. Both spatial and spectral basis functions are well known functions, thus the developed analytical model can be used to easily examine the HRTF data feature-individualization. An important finding of this thesis is that the HRTF decomposition with the spatial basis functions can be well approximated by a finite number, which is defined as the HRTF spatial dimensionality. The dimensionality determines the least number of the HRTF measurements in space. We perform high resolution HRTF measurements on a KEMAR mannequin in a semi-anechoic acoustic chamber. Both signal processing aspects to extract HRTFs from the raw measurements and a practical high resolution spatial sampling scheme have been given in this thesis.