We demonstrate far-field fluorescence microscopy with a focal-plane resolution of 15-20 nm in biological samples. The 10- to 12-fold multilateral increase in resolution below the diffraction barrier has been enabled by the elimination of molecular triplet state excitation as a major source of photobleaching of a number of dyes in stimulated emission depletion microscopy. Allowing for relaxation of the triplet state between subsequent excitation-depletion cycles yields an up to 30-fold increase in total fluorescence signal as compared with reported stimulated emission depletion illumination schemes. Moreover, it enables the reduction of the effective focal spot area by up to approximately 140-fold below that given by diffraction. Triplet-state relaxation can be realized either by reducing the repetition rate of pulsed lasers or by increasing the scanning speed such that the build-up of the triplet state is effectively prevented. This resolution in immunofluorescence imaging is evidenced by revealing nanoscale protein patterns on endosomes, the punctuated structures of intermediate filaments in neurons, and nuclear protein speckles in mammalian cells with conventional optics. The reported performance of diffraction-unlimited fluorescence microscopy opens up a pathway for addressing fundamental problems in the life sciences.