  The magnetic penetration depth $\lambda (T)$ and the upper critical field $%
\mu_{0}H_{c2}(T_{c})$ of the non-centrosymmetric (NCS) superconductor Y$_{2}
$C$_{3}$ have been measured using a tunnel-diode (TDO) based resonant
oscillation technique. We found that the penetration depth $\lambda (T)$ and
its corresponding superfluid density $\rho_{s}(T)$ show linear temperature
dependence at very low temperatures ($T\ll T_{c}$), indicating the existence of
line nodes in the superconducting energy gap. Moreover, the upper critical
field $\mu_{0}H_{c2}(T_{c})$ presents an upturn at low temperatures with a
rather high value of $\mu_{0}H_{c2}(0)$ $\simeq 29$T, which slightly exceeds
the weak-coupling Pauli limit. We discuss the possible origins for these
nontrivial superconducting properties, and argue that the nodal gap structure
in Y$_{2}$C$_{3}$ is likely attributed to the absence of inversion symmetry,
which allows the admixture of spin-singlet and spin-triplet pairing states.
