No one has mentioned Neon Genesis Evangelion, and that makes me upset!
Robot mechs* versus aliens** in a post-apocalyptic world. Robots are only able to be piloted by children because they're more easily manipulated and adaptive the the neuron system in place of the Evangelions (the mechs). Less focus on the robots and action and more on the characters themselves, and deeply delves into the psychological states of the pilots - and, despite the overall dark moods, the theme itself is quite heartwarming and inspiring, that all people have the power within themselves to come back from the brink of depression and to conquer their insecurities.
*not actually robot mechs
**not actually aliens
