(Abridged) We present new K-band spectroscopy for a sample of 48 starburst galaxies, obtained using UKIRT in Hawaii. This constitutes a fair sample of the most common types of starburst galaxies found in the nearby Universe. The variety of near infrared spectral features shown by these galaxies implies different bursts characteristics, which suggests that we survey galaxies with different star formation histories or at different stages of their burst evolution. Using synthetic starburst models, we conclude that the best ensemble of parameters which describe starburst galaxies in the nearby universe are a constant rate of star formation, a Salpeter IMF with an upper mass cutoff equal to 30 solar mass and bursts ages between 10 Myr and 1 Gyr. The model is fully consistent with the differences observed in the optical and FIR between the different types of starbursts. It suggests that HII galaxies have younger bursts and lower metallicities than SBNGs, while LIRGs have younger bursts but higher metallicities. Our observations suggest that the starburst phenomenon must be a sustained or self--sustained phenomenon: either star formation is continuous in time or multiple bursts happen in sequence over a relatively long period of time. The generality of our observations implies that this is a characteristic of starburst galaxies in the nearby Universe.
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