  We report the discovery of superconductivity in Nb$_3$Pd$_{x}$Se$_7$ with a
$x$-dependent superconducting transition-temperature as high as $T_c \simeq 2.1
$ K for $x \simeq0.7$ (middle point of the resistive transition). Needle-like
single crystals display anisotropic upper-critical fields with an anisotropy
$\gamma = H^{b}_{c2}/H^{a}_{c2}$ as large as 6 between fields applied along
their needle axis (or $b-$axis) or along the $a-$axis. As for the Fe based
superconductors $\gamma$ is temperature-dependent suggesting that
Nb$_3$Pd$_{0.7}$Se$_7$ is a multi-band superconductor. This is supported by
band structure calculations which reveal a Fermi surface composed of
quasi-one-dimensional and quasi-two-dimensional sheets of hole character, as
well as three-dimensional sheets of both hole- and electron-character.
Remarkably, $H^{b}_{c2}$ is observed to saturate at $H^{b}_{c2}(T \rightarrow 0
\text{K}) \simeq 14.1$ T which is $4.26 \times H_p$ where $H_p$ is the
Pauli-limiting field in the weak-coupling regime. The synthesis procedure
yields additional crystals belonging to the Nb$_2$Pd$_{x}$Se$_5$ phase which
also becomes superconducting when the fraction of Pd is varied. For both phases
we find that superconductivity condenses out of an anomalous metallic state,
i.e. displaying $\partial \rho/ \partial T < 0$ above $T_c$ similarly to what
is observed in the pseudogap-phase of the underdoped cuprates. An anomalous
metallic state, low-dimensionality, multi-band character, extremely high and
anisotropic $H_{c2}$s, are all ingredients for unconventional
superconductivity.
