The majority of studies of the living cell rely on capturing images using fluorescence microscopy. Unfortunately, for centuries, diffraction of light was limiting the spatial resolution in the optical microscope: structural and molecular details much finer than about half the wavelength of visible light (~200 nm) could not be visualized, imposing significant limitations on this otherwise so promising method. The surpassing of this resolution limit in far-field microscopy is currently one of the most momentous developments for studying the living cell, as the move from microscopy to super-resolution microscopy or 'nanoscopy' offers opportunities to study problems in biophysical and biomedical research at a new level of detail. This review describes the principles and modalities of present fluorescence nanoscopes, as well as their potential for biophysical and cellular experiments. All the existing nanoscopy variants separate neighboring features by transiently preparing their fluorescent molecules in states of different emission characteristics in order to make the features discernible. Usually these are fluorescent 'on' and 'off' states causing the adjacent molecules to emit sequentially in time. Each of the variants can in principle reach molecular spatial resolution and has its own advantages and disadvantages. Some require specific transitions and states that can be found only in certain fluorophore subfamilies, such as photoswitchable fluorophores, while other variants can be realized with standard fluorescent labels. Similar to conventional far-field microscopy, nanoscopy can be utilized for dynamical, multi-color and three-dimensional imaging of fixed and live cells, tissues or organisms. Lens-based fluorescence nanoscopy is poised for a high impact on future developments in the life sciences, with the potential to help solve long-standing quests in different areas of scientific research.