We present results from the first population synthesis study of protostellar discs. We analyse the evolution and properties of a large sample of protostellar discs formed in a radiation hydrodynamical simulation of star cluster formation. Due to the chaotic nature of the star formation process, we find an enormous diversity of young protostellar discs, including misaligned discs, and discs whose orientations vary with time. Star-disc interactions truncate discs and produce multiple systems. Discs may be destroyed in dynamical encounters and/or through ram-pressure stripping, but reform by later gas accretion. We quantify the distributions of disc mass and radii for protostellar ages up to $approx 10^5$ yrs. For low-mass protostars, disc masses tend to increase with both age and protostellar mass. Disc radii range from of order ten to a few hundred au, grow in size on timescales $le 10^4$ yr, and are smaller around lower-mass protostars. The radial surface density profiles of isolated protostellar discs are flatter than the minimum mass solar nebula model, typically scaling as $Sigma propto r^{-1}$. Disc to protostar mass ratios rarely exceed two, with a typical range of $M_{rm d}/M_* = 0.1-1$ to ages $le 10^4$ yrs and decreasing thereafter. We quantify the relative orientation angles of circumstellar discs and the orbit of bound pairs of protostars, finding a preference for alignment that strengths with decreasing separation. We also investigate how the orientations of the outer parts of discs differ from the protostellar and inner disc spins for isolated protostars and pairs.
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