  We study the usual one-dimensional Kondo lattice model (1D KLM) using the
non-Abelian bosonization formalism. At half-filling, we obtain a Kondo
insulator with a gap in both charge and spin excitations which varies quite
linearly with the Kondo exchange $J_K$. It consists of a Spin Density Glass
state, or a $q=\pi$ spin density wave weakly pinned by a nearly
antiferromagnetically ordered spin array. We will analyze the stability of the
SDG state in presence of randomness. We will compare these results with those
obtained in the one-dimensional Heisenberg-Kondo lattice model (1D HKLM) where
the spins are coupled through a large Heisenberg exchange. Away from
half-filling, the system is metallic and yields a very small spin gap which is
equal to the one-impurity Kondo gap $T_k^{(imp)}$. Unlike the one-impurity
Kondo model, we will show why this Kondo phase cannot rule the fixed point of
the 1D KLM, away from half-filling. We will rather obtain a normal
heavy-fermion state controlled by the energy scale $T_{coh}\sim
(T_k^{(imp)})^2/t$ (t is the hopping term) with a quite long-range
antiferromagnetic polarization. It is important to notice that both the SDG
state and the normal heavy-fermion state are susceptible to occur near a
magnetic instability.
