Online:Twitch

Twitch integration for Elder Scrolls Online was announced on May 16, 2018. It allows you to obtain promotional in-game rewards for using Twitch.

In order to receive the promotional rewards, you must first go to account.elderscrollsonline.com and link your ESO account with your Twitch account. Once this is done, any rewards you receive will be delivered in-game.  Account linking is currently only available for PC/Mac.

Twitch Drops
Twitch Drops is a major promotion for ESO, offering in-game rewards for watching the game on Twitch. The promotion went live on May 22, 2018.

To receive drops, you must watch certain "supported streams" on Twitch. If you receive a drop, you will be notified via email (it will say "code redeemed"; you do not need to do anything with this code as the crate has been added to your account.)  Depending on how drops are set up for that particular stream, it may be during the stream or after it ends. 

Only one drop is available: an Ouroboros Crown Crate. See the Crate article for more information on the potential rewards you can receive when opening one of these Crates.

Extensions

Extensions are tools that Twitch streamers can apply to their streams. Bethesda has created their own which is enabled during major streaming events such as expo presentations. In addition to various interactive functions made available during the stream, watching with the extension enabled also often provides promotional in-game rewards for viewers. This requires viewers to link their ESO account to Twitch; Bethesda.net accounts can also be linked to receive rewards for other games such as Legends.

To ensure that the extension is enabled, you can click the puzzle piece icon in the bottom right corner of the stream. Note that extensions may not be supported on all platforms; it is recommended to view it on PC.

To date, there have been several promotional rewards offered through the Twitch extension for various events:
E3 2018: Ice Wolf mount and 5 Ouroboros Crown Crates
QuakeCon 2018: Flame Atronach Wolf pet and 5 Ouroboros Crown Crates
QuakeCon 2019: Flame Atronach Senche-Jaguar pet and an Ouroboros Crown Crate
QuakeCon 2020: Flame Atronach Pony pet and an Ouroboros Crown Crate
QuakeCon 2021: Flame Atronach Wolf pet and an Ouroboros Crown Crate

Games on Twitch
Beginning on May 21, 2018, ESO products were available for purchase through the Games on Twitch online store. Specifically, the following products were available:
ESO Summerset Standard Edition
Crown Packs
ESO Plus subscription time
Making any ESO purchase through Twitch would reward you with a "Twitch Crate" containing various digital items for your Twitch account as well as an Ouroboros Crown Crate. However, ESO Plus was significantly more expensive to buy through Twitch for customers located outside the US.

The Games on Twitch service was discontinued on November 27, 2018.

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See Also
Twitch - The Elder Scrolls Online — A list of current ESO livestreams.
Elder Scrolls Online - Twitch Drops & More Are Coming To The Elder Scrolls Online!
Announcing Our #BE3 Twitch Extension
Legends Twitch Drops
Daily Rewards — another way to receive rewards
ESO Live — the official ESO livestream, with regular Twitch integration