  We establish the large deviation principle for the slow variables in
slow-fast dynamical system driven by both Brownian noises and L\'evy noises.
The fast variables evolve at much faster time scale than the slow variables,
but they are fully inter-dependent. We study the asymptotics of the logarithmic
functionals of the slow variables in the three regimes based on viscosity
solutions to the Cauchy problem for a sequence of partial integro-differential
equations. We also verify the comparison principle for the related Cauchy
problem to show the existence and uniqueness of the limit for viscosity
solutions.
