A non-spoilery overview of the series now that I've finished it. No spoilers in comments, please, for the benefit of those who haven't seen it; there's a spoiler-protected post just before this one.
Vision of Escaflowne is a 26-episode anime that originally aired in 1996. It was cut from 39 episodes in pre-production, and the resulting compression means it's very fast-paced—sometimes too much so for me, especially in the ending. Its protagonist is Hitomi, a Japanese schoolgirl who is pulled into another world, where plot ensues. On one hand, she has Magical Mystical powers like predicting the future with Tarot cards, but on the other, she's a lean-figured track star and generally pretty sensible (except, alas, when the plot requires her to be otherwise).
I believe this has anime tropes like whoa: fate, fortune-telling, and destiny-changing; crossing between worlds; dragons; wingèd people and significant falling feathers; giant flying robot suits of armor; animal-people; missing family members; secret family members; prosthetics; shapechanging; ghosts, spirits, or visions of dead people; and rainy days of doom. There's a lot of fighting, and a lot of relationship stuff, and because I wasn't as drawn to the characters as a lot of people seem to be, my occasional chant was "More giant robots! Less teenage angst!"
That's just personal preference: to its credit, the series manages to give depth to a great many of the characters, making them more complicated than they first appear. I do appreciate that, and as I said, most people seem to like the characters more than I do.
If the characters and relationships had really pulled me in, I might have been able to deal with the whole fate thing, which is the core of the plot. Instead, I had two problems with it. First, the series' concept of fate seemed kind of woo-woo New Agey to me, which is not my kind of thing. Second, perhaps because of the plot compression, it often didn't seem to make sense even on its own terms. Amusing, in its pseudoscientific babble, but not really comprehensible.
So, I think this was a good educational experience for me, as an older series with the anime tropes like whoa, but I didn't really warm to it. However, the reasons I didn't warm to it are certainly not universal, and those already interested in anime might check out the first four or five episodes.