  Drawing inspiration from Nyberg's paper~\cite{Nyb91} on perfect nonlinearity
and the $c$-differential notion we defined in~\cite{EFRST20}, in this paper we
introduce the concept of $c$-differential bent functions in two different ways
(thus extending Kumar et al.~\cite{Ku85} classical definition). We further
extend the notion of perfect $c$-nonlinear introduced in~\cite{EFRST20}, also
in two different ways, and show that, in both cases, the concepts of
$c$-differential bent and perfect $c$-nonlinear are equivalent (under some
natural restriction of the parameters). Some constructions of functions with
these properties are also provided; one such construction provides a large
class of PcN functions with respect to all $c$ in some subfield of the field
under consideration. We also show that both our classes of $0$-differential
bents are supersets of permutation polynomials, and that Maiorana-McFarland
bent functions are not differential bent (of the first kind).
