Gilgamesh is... hard to sum up. It's also hard to explain exactly what it is I like about it. But what it essentially boils down to is the story, Gilgamesh a very intricate, multifaceted plot that requires you tune in and pay attention, which can be difficult because, and I say this acknowledging it as one of my favorite anime; Gilgamesh is really, really boring.
It might not look boring at first. But...
Gilgamesh takes place after a cataclysmic event clouded the world's atmosphere with an electromagnetic pulse, rendering all digital devices useless and replacing the once blue sky with a shimmering mirror, an event known as XX due to it taking place on October 10th. And the two leads; Kiyoko and Tatsuya Modoka are adopted by a wealthy woman to combat a group of people with supernatural powers known as Gilgamesh. And the reason why is because Tatsuya, along with several other unique teenagers, have a mysterious power called Dynamis. That's the short summary, if I were to go into all of the plot elements it would take another five paragraphs. And when I said the show was boring, I meant the setting is drab, the characters, while lively and varied, feel very understated, and the few action scenes feel half-hearted and dull. The story is what has to draw you in, and that's exactly what I like about it.
For one, the two main leads are very compelling, with their father being a much more important part of the plot that practically strong arms them into being part of the central conflict. There's also a dichotomy between the two with Kyoko being born before XX and Tatsuya being born after, that as a big impact on the plot as well. Also the art direction is beautiful, and when coupled with the excellent soundtrack helps create a fantastic atmosphere of dread and despair, which is important given the state of the world and what happens when the final episode rolls around. In addition, the voice-acting is superb, with Shelly Calene Black stealing the show as Kyoko. And perhaps most importantly, everything has a bearing on the central plot. And the longer it continues, the more fascinating it becomes.
Pretty much sums it up.
Gilgamesh definitely requires patience, and if you cant get invested in the plot, you wont get invested at all. That said, the story is among the most compelling I've seen in an anime to date, and that helps make up for the overall drab feel of the series (and the “emo” character designs which apparently bothers people). The ending feels a bit too busy, it probably could have stood to be a two parter, but it's also one of the most fascinating anime I've ever seen.