  We study the connection of the dynamics in relativistic field theories in a
strong magnetic field with the dynamics of noncommutative field theories
(NCFT). As an example, the Nambu-Jona-Lasinio models in spatial dimensions $d
\geq 2$ are considered. We show that this connection is rather sophisticated.
In fact, the corresponding NCFT are different from the conventional ones
considered in the literature. In particular, the UV/IR mixing is absent in
these theories. The reason of that is an inner structure (i.e., dynamical
form-factors) of neutral composites which plays an important role in providing
consistency of the NCFT. An especially interesting case is that for a magnetic
field configuration with the maximal number of independent nonzero tensor
components. In that case, we show that the NCFT are finite for even $d$ and
their dynamics is quasi-(1+1)-dimensional for odd $d$. For even $d$, the NCFT
describe a confinement dynamics of charged particles. The difference between
the dynamics in strong magnetic backgrounds in field theories and that in
string theories is briefly discussed.
