As　an　effective　technique　to　relieve　the　problem　of　resource　constraints　on　mobile　devices　(MDs),　the　computation　offloading　utilizes　powerful　cloud　and　edge　resources　to　process　the　computation-intensive　tasks　of　mobile　applications　uploaded　from　MDs.　In　cloud-edge　computing,　the　resources　(e.g.,　cloud　and　edge　servers)　that　can　be　accessed　by　mobile　applications　may　change　dynamically.　Meanwhile,　the　parallel　tasks　in　mobile　applications　may　lead　to　the　huge　solution　space　of　offloading　decisions.　Therefore,　it　is　challenging　to　determine　proper　offloading　plans　in　response　to　such　high　dynamics　and　complexity　in　cloud-edge　environments.　The　existing　studies　often　preset　the　priority　of　parallel　tasks　to　simplify　the　solution　space　of　offloading　decisions,　and　thus　the　proper　offloading　plans　cannot　be　found　in　many　cases.　To　address　this　challenge,　we　propose　a　novel　real-time　and　Dependency-aware　task　Offloading　method　with　Deep　Q-networks　(DODQ)　in　cloud-edge　computing.　In　DODQ,　mobile　applications　are　first　modeled　as　Directed　Acyclic　Graphs　(DAGs).　Next,　the　Deep　Q-Networks　(DQN)　is　customized　to　train　the　decision-making　model　of　task　offloading,　aiming　to　quickly　complete　the　decision-making　process　and　generate　new　offloading　plans　when　the　environments　change,　which　considers　the　parallelism　of　tasks　without　presetting　the　task　priority　when　scheduling　tasks.　Simulation　results　show　that　the　DODQ　can　well　adapt　to　different　environments　and　efficiently　make　offloading　decisions.　Moreover,　the　DODQ　outperforms　the　state-of-art　methods　and　quickly　reaches　the　optimal/near-optimal　performance.