  In this review article, we discuss the distinction and possible equivalence
between scale invariance and conformal invariance in relativistic quantum field
theories. Under some technical assumptions, we can prove that scale invariant
quantum field theories in $d=2$ dimension necessarily possess the enhanced
conformal symmetry. The use of the conformal symmetry is well appreciated in
the literature, but the fact that all the scale invariant phenomena in $d=2$
dimension enjoy the conformal property relies on the deep structure of the
renormalization group. The outstanding question is whether this feature is
specific to $d=2$ dimension or it holds in higher dimensions, too. As of
January 2014, our consensus is that there is no known example of scale
invariant but non-conformal field theories in $d=4$ dimension under the
assumptions of (1) unitarity, (2) Poincar\'e invariance (causality), (3)
discrete spectrum in scaling dimension, (4) existence of scale current and (5)
unbroken scale invariance in the vacuum. We have a perturbative proof of the
enhancement of conformal invariance from scale invariance based on the higher
dimensional analogue of Zamolodchikov's $c$-theorem, but the non-perturbative
proof is yet to come. As a reference we have tried to collect as many
interesting examples of scale invariance in relativistic quantum field theories
as possible in this article. We give a complementary holographic argument based
on the energy-condition of the gravitational system and the space-time
diffeomorphism in order to support the claim of the symmetry enhancement. We
believe that the possible enhancement of conformal invariance from scale
invariance reveals the sublime nature of the renormalization group and
space-time with holography.
