Metasurfaces offer a unique platform to precisely control optical wavefronts and enable the realization of flat lenses, or metalenses, which have the potential to substantially reduce the size and complexity of imaging systems and to realize new imaging modalities. However, it is a major challenge to create achromatic metalenses that produce a single focal length over a broad wavelength range because of the difficulty in simultaneously engineering phase profiles at distinct wavelengths on a single metasurface. For practical applications, there is a further challenge to create broadband achromatic metalenses that work in the transmission mode for incident light waves with any arbitrary polarization state. We developed a design methodology and created libraries of meta-units-building blocks of metasurfaces-with complex cross-sectional geometries to provide diverse phase dispersions (phase as a function of wavelength), which is crucial for creating broadband achromatic metalenses. We elucidated the fundamental limitations of achromatic metalens performance by deriving mathematical equations that govern the tradeoffs between phase dispersion and achievable lens parameters, including the lens diameter, numerical aperture (NA), and bandwidth of achromatic operation. We experimentally demonstrated several dielectric achromatic metalenses reaching the fundamental limitations. These metalenses work in the transmission mode with polarization-independent focusing efficiencies up to 50% and continuously provide a near-constant focal length over λ = 1200-1650 nm. These unprecedented properties represent a major advance compared to the state of the art and a major step toward practical implementations of metalenses.