It's a matter of semantics with what people refer to my genre of choice as, or whether they consider it one, but mine is based on a pre-internet label popularized in print-magazines that I still hang onto.
Action/RPG or ARPG for short. My personal definition -
Making the rogue-like genre a little more focused by removing randomization and adding goals as well as a lot more forgiveness, Zelda is the main progenitor, but even Zelda has it's influences.
It's a subgenre defined by, but not limited to, elements such as top-down view, a connected overworld leading to labyrinthine dungeons and challenges, hopefully environment based and lateral thought puzzles, and a sense of character building and progression without traditional stat tracking, dice rolls, and menu based combat. The world and solutions tend to open up to you via items/abilities you acquire and by things you've learned and remembered to backtrack to.
It's rooted in high fantasy, but does not have to be. The genre also evolved into sidescrolling 2D, and 3D, but has slowly lost it's prominence and familiarity as almost all games merge genres now.
My examples -
Top Down:
Zelda NES, A Link to the Past, Link's Awakening, Oracle of Seasons/Ages, Minish Cap etc
Chrystallis
Beyond Oasis & Legend of Oasis
Alundra
Illusion of Gaia
Crusader of Centy
Twisted Tales of Spike McFang
Blood Omen: Legacy of Kain
Herc's Adventure
Cladun & Cladun x2
YSI & II
2D sidescrolling:
Zelda II
Faxanadu
Castlevania: Simon's Quest
Clash at Demonhead
YSIII
Popfulmail
Castlevania: Symphony of the Night (and all those GBA & DS games)
Isometric:
Solstice
Landstalker
Dark Savior
Vagrant Story (leniency given here)
Bastion (a really modern one that shows how the interwoven world design and puzzle aspect is really fading away)
3D:
Zelda: Ocarina of Time, Wind Waker, Twilight Princess & Skyward Sword
Legacy of Kain: Soul Reaver
Ico
Shadow of the Colossus (in this instance, Zelda's dungeons were mobile, the bosses were on the outside of them)
...most of the YS series and games like Shining Force EXA/Neo are close, but have gone the action/adventure route, or drifted more into Secret of Mana/Evermore or Diablo territory. I feel like they're inseparable sister cities, though. I feel like Dark Souls & Demon's Souls, aside from Zelda, are the closest thing there is to a modern evolution of this subgenre.