A distribution function $F$ on the nonnegative halfline is called subexponential if $\lim_{x\to \infty}(1-F^{*n}(x))/(1-F(x))=n$ for all $n\geq 2$. We obtain new sufficient conditions for subexponential distributions and related classes of distribution functions. Our results are formulated in terms of the hazard rate. We also analyse the rate of convergence in the definition and discuss the asymptotic behaviour of the remainder term $R_n(x)=1-F^{*n}(x)-n(1-F(x))$. We use the results in studying subordinated distributions and we conclude the paper with some multivariate extensions of our results.