  We study the competition between magnetic and spin-liquid phases in the
Hubbard model on the anisotropic triangular lattice, which is described by two
hopping parameters t and t' in different spatial directions and is relevant for
layered organic charge-transfer salts. By using a variational approach that
includes spiral magnetic order, we provide solid evidence that a spin-liquid
phase is stabilized in the strongly-correlated regime and close to the
isotropic limit t'/t=1. Otherwise, a magnetically ordered spiral state is
found, connecting the (collinear) N\'eel and the (coplanar) 120{\deg} phases.
The pitch vector of the spiral phase obtained from the unrestricted
Hartree-Fock approximation is substantially renormalized in presence of
electronic correlations, and the N\'eel phase is stabilized in a wide regime of
the phase diagram, i.e., for t'/t < 0.75. We discuss these results in the
context of organic charge-transfer salts.
