We consider the effect of the curvature in fermionic dispersion on the observable properties of Luttinger liquid (LL). We use the bosonization technique where the curvature is irrelevant perturbation, describing the decay of LL bosons (plasmon modes). When possible, we establish the correspondence between the bosonization and the fermionic approach. We analyze modifications in density correlation functions due to curvature at finite temperatures, T. The most important application of our approach is the analysis of the Coulomb drag by small momentum transfer between two LLs, which is only possible due to curvature. Analyzing the ac transconductivity in the one-dimensional drag setup, we confirm the results by Pustilnik for T dependence of drag resistivity, R12 ∼ T2 at high and R12 ∼ T5 at low temperatures. The bosonization allows for treating both intra- and interwire electron-electron interactions in all orders, and we calculate exact prefactors in the low- T drag regime. The crossover temperature between the two regimes is T1 ∼ EF Δ, with Δ relative difference in plasmon velocities. We show that Δ≠0 even for identical wires, due to lifting of degeneracy by interwire interaction, U12, leading to crossover from R12 ∼ U122 T2 to R12 ∼ T5 / U12 at T∼ U12.