Diffusion MRI (dMRI) is a popular noninvasive imaging modality for the investigation of the neonate brain. It enables the assessment of white matter integrity, and is particularly suited for studying white matter maturation in the preterm and term neonate brain. Diffusion tractography allows the delineation of white matter pathways and assessment of connectivity in vivo. In this review, we address the challenges of performing and analysing neonate dMRI. Of particular importance in dMRI analysis is adequate data preprocessing to reduce image distortions inherent to the acquisition technique, as well as artefacts caused by head movement. We present a summary of techniques that should be used in the preprocessing of neonate dMRI data, and demonstrate the effect of these important correction steps. Furthermore, we give an overview of available analysis techniques, ranging from voxel-based analysis of anisotropy metrics including tract-based spatial statistics (TBSS) to recently developed methods of statistical analysis addressing issues of resolving complex white matter architecture. We highlight the importance of resolving crossing fibres for tractography and outline several tractography-based techniques, including connectivity-based segmentation, the connectome and tractography mapping. These techniques provide powerful tools for the investigation of brain development and maturation.