Weighted inequalities which widely generalize the Turán-Kubilius inequality are established. The following is typical: Let w(m) be a non- negative real-valued arithmetic function which satisfies w(q) \(\ll 1\), w(qm) \(\ll w(q)w(m)\) uniformly for prime-powers q and positive integers m, \((q,m)=1\). Suppose that for some \(\beta >1\) the sums \(y^{-1}\sum _{n\leq y}w(n)^{\beta},\quad y\geq 1\) are uniformly bounded. If \(\alpha\) \(\geq 2\), then \[ ((1/x)\sum _{n\leq x}w(n)\quad | f(n)-\sum _{q\leq x}f(q)/q| ^{\alpha})^{1/\alpha}\quad \ll \] \[ \ll \quad (\sum _{q\leq x}| f(q)| ^ 2/q)^{1/2}\quad +\quad (\sum _{q\leq x}| f(q)| ^{\alpha}/q)^{1/\alpha} \] uniformly for additive functions f, real \(x\geq 1\). If \(00}\{(\sum _{q\leq x,\quad | f(q)| \leq \eta}| f(q)| ^ 2/q)^{1/2}\quad +\quad (\sum _{q\leq x,\quad | f(q)| >\eta}| f(q)| ^{\alpha}/q)^{1/\alpha}\}. \] The method of proof is based on the author's earlier paper [Can. J. Math. 32, 893-907 (1980; Zbl 0444.10051)], and quite different from that of P. Turán and I. Kubilius [see \textit{I. Kubilius}, Probabilistic methods in number theory (Russian) (1959; Zbl 0092.044)]. Some functional analytic background is given to show that the form of these inequalities is appropriate, the relation to previous results discussed, and a number of applications indicated.