  We study one dimensional fermionic and bosonic gases with repulsive power-law
interactions $1/|x|^{\beta}$, with $\beta>1$, in the framework of
Tomonaga-Luttinger liquid (LL) theory. We obtain an accurate analytical
expression linking the LL parameter to the microscopic Hamiltonian, for
arbitrary $\beta$ and strength of the interactions. In the presence of a small
periodic potential, power-law interactions make the LL unstable towards the
formation of a cascade of lattice solids with fractional filling, thus forming
a "Luttinger staircase". Several of these quantum phases and phase transitions
are realized with groundstate polar molecules and weakly-bound magnetic
Feshbach molecules.
