In this paper, we study analogues of the law of the iterated logarithm for delayed sums of independent random variables. In the i.i.d. case, necessary and sufficient conditions for such analogues are obtained. We apply our results to find convergence rates for expressions of the form $P\lbrack |S_n| > b_n \rbrack$ and $P\lbrack \sup_{k\geqq n} |S_k/b_k| > \varepsilon \rbrack$ for certain upper-class sequences $(b_n)$. In this connection, certain theorems of Erdos, Baum and Katz are also generalized.