Necessary and sufficient conditions are obtained for the weak convergence of random sums and maximum random sums of i.i.d. random variables. Limit distributions for these sums are described. Let \(v\), \(N_n\), \(X_n\), \(n=1,2,\dots\), be random variables defined on a probability space \((\Omega,{\mathcal A},P)\), where \(v\) is positive with probability one, each \(N_n\) takes natural values, and \(X_n\), \(n=1,2,\dots\), are i.i.d. with \(EX_k=a\neq 0\) and \(\text{Var }X_k= \sigma^2\), \(0d>>F(x)\) for some distribution function \(F\) it is necessary and sufficient that \[ P\Biggl({S_{[vn]}- avn\over\sqrt{\sigma^2n}}+ {a(N_n- vn)\over\sqrt{\sigma^2n}}d>> F(x).\tag{1} \] Theorem 2. Assume that \(EX_k=a>0\). In order that \(P(\overline Z_nd>>F(x)\) for some distribution function \(F\) it is necessary and sufficient that (1) holds.