  We consider a one-dimensional system of interacting bosons in a random
potential. At zero temperature, it can be either in the superfluid or in the
insulating phase. We study the transition at weak disorder and moderate
interaction. Using a systematic approach, we derive the renormalization group
equations at two-loop order and discuss the phase diagram. We find the
universal form of the correlation functions at the transitions and compute the
logarithmic corrections to the main universal power-law behavior. In order to
mimic large density fluctuations on a single site, we study a simplified model
of disordered two-leg bosonic ladders with correlated disorder across the rung.
Contrarily to the single-chain case, the latter system exhibits a transition
between a superfluid and a localized phase where the exponents of the
correlation functions at the transition do not take universal values.
