Google has announced that support for the Chromecast (1st generation-2013) is officially ending, which means the company will no longer offer updates. A message announcing the end of support has appeared on several Google support pages.

The device will no longer receive software updates or security patches. The nearly 10 years of support shows that the device has had a very good run. This is not uncommon in consumer electronics, especially when a product costs just $35. Google warns that end users “may notice a degradation in performance.”
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Essentially, it seems that 1st generation Chromecasts won't stop working immediately, but their functionality is likely to be affected over time as the ecosystem around them updates and progresses.
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The latest firmware for the 2013 Chromecast is 1.36.159268, which was released in November 2022. As the company had said, this version brought some bug fixes and improvements.

On the support page (last updated on April 27, 2023), Google says:
“Support for Chromecast (1st generation) has ended, meaning these devices no longer receive software or security , and Google no longer provides technical support for them. Users may experience performance degradation.“.
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The original Chromecast was a small and surprisingly popular device, shaped like a dongle, designed to plug into an HDMI port on TV so you could cast video content from your phone, tablet, or laptop to the big screen.
