  In one-dimensional quantum wires the interplay of electron correlations and
impurities strongly influences the low-energy physics. The diversity of energy
scales and the competition of correlations in interacting Fermi systems can be
treated very efficiently with the functional renormalization group (fRG),
describing the gradual evolution from a microscopic model Hamiltonian to the
effective low-energy action as a function of a continuously decreasing energy
cutoff. The fRG provides the universal low-energy asymptotics as well as
nonuniversal properties, and in particular an answer to the important question
at what scale the ultimate asymptotics sets in.
  We investigate the influence of impurities on spectral and transport
properties of fermionic lattice models with short-range interactions. The
results capture relevant energy scales and crossover phenomena, in addition to
the universal low-energy asymptotics. For weak and intermediate impurity
strengths the asymptotic behavior is approached only at rather low energy
scales, accessible only for very large systems. For spin-1/2 systems
two-particle backscattering leads to striking effects, which are not captured
if the bulk system is approximated by its low-energy fixed point, the Luttinger
model. In particular, the expected decrease of spectral weight near the
impurity and of the conductance at low energy scales is often preceded by a
pronounced increase, and the asymptotic power laws are modified by logarithmic
corrections.
