The human microbiome plays an important and increasingly recognized role in human health. Studies of the microbiome typically use targeted sequencing of the 16S rRNA gene, whole metagenome shotgun sequencing, or other meta-omic technologies to characterize the microbiome's composition, activity, and dynamics. Processing, analyzing, and interpreting these data involve numerous computational tools that aim to filter, cluster, annotate, and quantify the obtained data and ultimately provide an accurate and interpretable profile of the microbiome's taxonomy, functional capacity, and behavior. These tools, however, are often limited in resolution and accuracy and may fail to capture many biologically and clinically relevant microbiome features, such as strain-level variation or nuanced functional response to perturbation. Over the past few years, extensive efforts have been invested toward addressing these challenges and developing novel computational methods for accurate and high-resolution characterization of microbiome data. These methods aim to quantify strain-level composition and variation, detect and characterize rare microbiome species, link specific genes to individual taxa, and more accurately characterize the functional capacity and dynamics of the microbiome. These methods and the ability to produce detailed and precise microbiome information are clearly essential for informing microbiome-based personalized therapies. In this review, we survey these methods, highlighting the challenges each method sets out to address and briefly describing methodological approaches.