Metasurfaces have the potential to miniaturize and improve the performance of any optical element, with applications spanning telecommunications, computing and wearable optics. However, the ability to retain functionality over a continuous, broad range of wavelengths is essential for their widespread adoption. So far, efforts to achieve this have been limited to either a reflection configuration or operation at wavelengths other than the visible. Here, we show that by judicious choice of nanostructures, one can independently control both the phase and group delay, leading to broadband transmissive achromatic metalenses and metalenses with tunable dispersion across the visible spectrum. We demonstrate diffraction-limited broadband focusing and achromatic imaging using metalenses across wavelengths from 470 nm to 670 nm, with focal length change of a few percent across this 200 nm bandwidth. This approach requires only a single layer metasurface whose thickness is on the wavelength scale, and does not involve spatial multiplexing or cascading.