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We present a very sensitive experimental setup for accurate tunable wavelength hyper-Rayleigh scattering (HRS) measurements of the molecular first hyperpolarizability β in the broad fundamental wavelength range of 600 to 1800 nm, allowing the wavelength-dependence of β, which is still not well-understood, to be studied in detail throughout and beyond resonance. An extensive set of wavelength-dependent HRS calibration data for various solvents is presented, which has also allowed us to successfully measure the β dispersion of prototypical dipolar and octupolar organic molecules (e.g. Disperse Red 1 (DR1), Crystal Violet (CV)) and organometallic complexes. This has resulted in extensive theoretical modeling enabling reliable derivation of β at other wavelengths and in particular extrapolation to the static limit, which in turn makes it possible to compare β between different molecules and to evaluate their potential for electro-optical (EO) applications.