We consider additive diophantine equations of degree $k$ in $s$ variables and establish that whenever $s\ge 3k+2$ then almost all such equations satisfy the Hasse principle. The equations that are soluble form a set of positive density, and among the soluble ones almost all equations admit a small solution. Our bound for the smallest solution is nearly best possible.

Comment: The results in this paper use an $L^2$-technique and supersede those in an earlier version (see arXiv:1110.3496) that relied on an $L^1$-argument, but for instructional purposes we found it useful to keep the older, technically simpler version. arXiv admin note: substantial text overlap with arXiv:1004.5527