In this work we examine the impact behavior of single and doubly curved structures with auxetic cores by using numerical methods. We have evaluated twelve different doubly and one single curved panels impacted by a spherical projectile at a velocity of 100 m/s. The results are firstly discussed by considering the effects of the curvature on the backplane displacement and the energy absorption of the panels. A selected toroidal configuration is then modeled as sandwich structure with an auxetic (re-entrant) core and a spherical projectile is shot on different locations from center to edge. The sandwich structure is made by carbon fiber/epoxy laminates, while the re-entrant core is represented with ABSplus plastic. It is found that the damaged area of the auxetic core differs from a traditional honeycomb under impact loading, and this is mainly due to its negative Poisson's ratio.