We describe the Survey for Pulsars and Extragalactic Radio Bursts (SUPERB), an ongoing pulsar and fast transient survey using the Parkes radio telescope. SUPERB involves real-time acceleration searches for pulsars and single-pulse searches for pulsars and fast radio bursts. We report on the observational setup, data analysis, multi-wavelength/messenger connections, survey sensitivities to pulsars and fast radio bursts and the impact of radio frequency interference. We further report on the first 10 pulsars discovered in the project. Among these is PSR~J1306$-$40, a millisecond pulsar in a binary system where it appears to be eclipsed for a large fraction of the orbit. PSR~J1421$-$4407 is another binary millisecond pulsar; its orbital period is $30.7$ days. This orbital period is in a range where only highly eccentric binaries are known, and expected by theory; despite this its orbit has an eccentricity of $10^{-5}$.
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