  We study the rich dynamics resulting from introducing static charged
particles (Wilson lines) in 2+1 and 3+1 dimensional gauge theories. Depending
on the charges of the external particles, there may be multiple defect fixed
points with interesting renormalization group flows connecting them, or an
exponentially large screening cloud can develop (defining a new emergent length
scale), screening the bare charge entirely or partially. We investigate several
examples where the dynamics can be solved in various weak coupling or double
scaling limits. Sometimes even the elementary Wilson lines, corresponding to
the lowest nontrivial charge, are screened. We consider Wilson lines in 3+1
dimensional gauge theories including massless scalar and fermionic QED$_4$, and
also in the ${\mathcal N}=4$ supersymmetric Yang-Mills theory. We also consider
Wilson lines in 2+1 dimensional conformal gauge theories such as QED$_3$ with
bosons or fermions, Chern-Simons-Matter theories, and the effective theory of
graphene. Our results in 2+1 dimensions have potential implications for
graphene, second-order superconducting phase transitions, etc. Finally, we
comment on magnetic line operators in 3+1 dimensions ('t Hooft lines) and argue
that our results for the infrared dynamics of electric and magnetic lines are
consistent with non-Abelian electric-magnetic duality.
