  In the present work we review the twisted field construction of quantum field
theory on noncommutative spacetimes based on twisted Poincar\'e invariance. We
present the latest development in the field, in particular the notion of
equivalence of such quantum field theories on a noncommutative spacetime, in
this regard we work out explicitly the inequivalence between twisted quantum
field theories on Moyal and Wick-Voros planes; the duality between deformations
of the multiplication map on the algebra of functions on spacetime
$\mathscr{F}(\mathbb{R}^4)$ and coproduct deformations of the Poincar\'e-Hopf
algebra $H\mathscr{P}$ acting on~$\mathscr{F}(\mathbb{R}^4)$; the appearance of
a nonassociative product on $\mathscr{F}(\mathbb{R}^4)$ when gauge fields are
also included in the picture. The last part of the manuscript is dedicated to
the phenomenology of noncommutative quantum field theories in the particular
approach adopted in this review. CPT violating processes, modification of
two-point temperature correlation function in CMB spectrum analysis and
Pauli-forbidden transition in ${\rm Be}^4$ are all effects which show up in
such a noncommutative setting. We review how they appear and in particular the
constraint we can infer from comparison between theoretical computations and
experimental bounds on such effects. The best bound we can get, coming from
Borexino experiment, is $\gtrsim 10^{24}$ TeV for the energy scale of
noncommutativity, which corresponds to a length scale $\lesssim 10^{-43}$ m.
This bound comes from a different model of spacetime deformation more adapted
to applications in atomic physics. It is thus model dependent even though
similar bounds are expected for the Moyal spacetime as well as argued
elsewhere.
