  The representation theories of the SU(2)$_k$-extended $N$=4 superconformal
algebras (SCAs) with $arbitrary$ level $k$ are developed being based on their
Feigin-Fuchs representations found recently by the present author. A basic unit
of the representation blocks consisting of eight \lq\lq boson-like\rq\rq\ and
eight \lq\lq fermion-like\rq\rq\ conformal fields is found to describe
arbitrary representations of the $N$=4 SU(2)$_k$ SCAs, including {\it unitary}
and {\it nonunitary} representations. The transformation properties of the
fundamental sets of the conformal fields under the $N$=4 SU(2)$_k$
superconformal symmetries are given. Then, the whole sets of the
charge-screening operators of the $N$=4 SU(2)$_k$ SCAs are identified out of
the sixteen conformal fields in the basic unit of the representation blocks.
The conditions for the {\it eligible} charge-screening operators are analyzed
in terms of the continuous parameters which enter in our vertex-operator forms
for the fundamental conformal fields of the representation blocks.
