We review the current status of various aspects of biopolymer translocation through nanopores and the challenges and opportunities it offers. Much of the interest generated by nanopores arises from their potential application to third-generation cheap and fast genome sequencing. Although the ultimate goal of single-nucleotide identification has not yet been reached, great advances have been made both from a fundamental and an applied point of view, particularly in controlling the translocation time, fabricating various kinds of synthetic pores or genetically engineering protein nanopores with tailored properties, and in devising methods (used separately or in combination) aimed at discriminating nucleotides based either on ionic or transverse electron currents, optical readout signatures, or on the capabilities of the cellular machinery. Recently, exciting new applications have emerged, for the detection of specific proteins and toxins (stochastic biosensors), and for the study of protein folding pathways and binding constants of protein–protein and protein–DNA complexes. The combined use of nanopores and advanced micromanipulation techniques involving optical/magnetic tweezers with high spatial resolution offers unique opportunities for improving the basic understanding of the physical behavior of biomolecules in confined geometries, with implications for the control of crucial biological processes such as protein import and protein denaturation. We highlight the key works in these areas along with future prospects. Finally, we review theoretical and simulation studies aimed at improving fundamental understanding of the complex microscopic mechanisms involved in the translocation process. Such understanding is a pre-requisite to fruitful application of nanopore technology in high-throughput devices for molecular biomedical diagnostics.