  Amorphous solids appear to react elastically to small external strains, but
in contrast to ideal elastic media, plastic responses abound immediately, at
any value of the strain. Such plastic responses are quasi-localized in nature,
with the ``cheapest" one being a quadrupolar source. The existence of such
plastic responses results in {\em screened elasticity} in which strains and
stresses can either quantitatively or qualitatively differ from the un-screened
theory, depending on the specific screening mechanism. Here we offer a theory
of such screening effects by plastic quadrupoles, dipoles and monopoles,
explain their natural appearance, and point out the analogy to electrostatic
screening by electric charges and dipoles. For low density of quadrupoles the
effect is to normalize the elastic moduli without a qualitative change compared
to pure elasticity theory; for higher density of quadrupoles the screening
effects result in qualitative changes. Predictions for the spatial dependence
of displacement fields caused by local sources of strains are provided and
compared to numerical simulations. We find that anomalous elasticity is richer
than electrostatics in having a screening mode that does not appear in the
electrostatic analog
