We present a complete survey of current star formation in the Perseus molecular cloud, made at 850 and 450 micron with SCUBA at the JCMT. Covering 3 deg^2, this submillimetre continuum survey for protostellar activity is second in size only to that of rho Ophiuchus (Johnstone et al. 2004). Complete above 0.4 msun (5 sigma detection in a 14 beam), we detect a total of 91 protostars and prestellar cores. Of these, 80% lie in clusters, representative of star formation across the Galaxy. Two of the groups of cores are associated with the young stellar clusters IC348 and NGC1333, and are consistent with a steady or reduced star formation rate in the last 0.5 Myr, but not an increasing one. In Perseus, 40--60% of cores are in small clusters (< 50 msun) and isolated objects, much more than the 10% suggested from infrared studies. Complementing the dust continuum, we present a C^18O map of the whole cloud at 1 resolution. The gas and dust show filamentary structure of the dense gas on large and small scales, with the high column density filaments breaking up into clusters of cores. The filament mass per unit length is 5--11 msun per 0.1 pc. Given these filament masses, there is no requirement for substantial large scale flows along or onto the filaments in order to gather sufficient material for star formation. We find that the probability of finding a submillimetre core is a strongly increasing function of column density, as measured by C^18O integrated intensity, prob(core) proportional to I^3.0. This power law relation holds down to low column density, suggesting that there is no A_v threshold for star formation in Perseus, unless all the low-A_v submm cores can be demonstrated to be older protostars which have begun to lose their natal molecular cloud.
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