Viruses alter host cell processes to optimize their replication cycle. Human adenoviruses (Ad) encode proteins that promote viral macromolecular synthesis and counteract innate and adaptive responses to infection. The focus of this review is on how Ad evades innate cellular responses to infection, including an interferon (IFN) response and a DNA damage response (DDR). Ad blocks the IFN response by inhibiting cytoplasmic signaling pathways and the activation of IFN-stimulated genes (ISGs), as well as the functions of ISG products, such as PML. Ad also inhibits DDR sensors, for instance, the Mre11-Rad50-Nbs1 complex, and DDR effectors like DNA ligase IV. These innate cellular responses impact many different viruses, and studies on Ad have provided broad insight into these areas.