A nanometre-scale pore in a solid-state membrane provides a new way of electronically probing the structure of single linear polymers, including those of biological interest in their native environments. Previous work with biological protein pores wide enough to let through and sense single-stranded DNA molecules demonstrates the power of using nanopores, but many future tasks; and applications call for a robust solid-state pore whose nanometre-scale dimensions and properties may be selected, as one selects the lenses of a microscope. Here we demonstrate a solid-state nanopore microscope capable of observing individual molecules of double-stranded DNA and their folding behaviour. We discuss extensions of the nanopore microscope concept to alternative probing mechanisms and applications, including the study of molecular structure and sequencing.