  We introduce a Lagrangian-space Effective Field Theory (LEFT) formalism for
the study of cosmological large scale structures. Unlike the previous
Eulerian-space construction, it is naturally formulated as an effective field
theory of extended objects in Lagrangian space. In LEFT the resulting finite
size effects are described using a multipole expansion parameterized by a set
of time dependent coefficients and organized in powers of the ratio of the
wavenumber of interest $k$ over the non-linear scale $k_{\rm NL}$. The
multipoles encode the effects of the short distance modes on the
long-wavelength universe and absorb UV divergences when present. There are no
IR divergences in LEFT. Some of the parameters that control the perturbative
approach are not assumed to be small and can be automatically resummed. We
present an illustrative one-loop calculation for a power law universe. We
describe the dynamics both at the level of the equations of motion and through
an action formalism.
