There are several reasons to be looking for a ROM:
But where can one find ROMs in the first place? What sites are likely to have the ROM I need?
This question is intended to be canonical; please add to the wiki answer instead of posting separate answers.
See also: How do I update Android on my device?
While they don't usually host ROMs directly, you may find that they have software useful for upgrading stock ROMs, such as Samsung's tool Kies. If you want to upgrade and stay stock then you should try official methods first.
Google hosts update packages for the flagship Nexus and Pixel devices at
https://android.clients.google.com/packages
You can't browse that link directly, but a web search is your friend here. Search for that link plus the code name of the device and the version you want; for example, searching android.clients.google.com/packages crespo ics results in a lot of pages with this link -- the OTA Ice Cream Sandwich update for the T-Mobile Nexus S. Links for various "factory images" for Nexus devices can be found here.
Probably the first place you should look after trying the official route is the XDA Developers' forums. Stock ROMs are usually posted in the General sub-forum for your device, and custom ROMs in the Android Development sub-forum for your device. No guarantees.
They don't have dedicated subforums for all devices, so if you have a less-popular device you should use the site search instead of the subforum search.
You may or may not want to check RootzWiki before XDA Forums above. Developers tend to move from forum to forum and RootzWiki provides devices for the best developers so a lot have jumped on board.
They have sections for many devices and for those that don't, there is a General section. The forums tend to be divided into 3 sections per device (not always): General, Development (ROMS and such), and Themes.
This site often has ROMs for more obscure devices, as well as popular ones.
This site provides a long list (about 800 pages) for all kinds of devices, including Android, Blackberry, Symbian, and more. It misses a filtering mechanism, unfortunately...
Grouped by vendor, then device. Seems to be quite up-to-date and quite complete -- for sure one of the best resources.
This site posts stock Samsung ROMs from all over. It does require you to sign up.
Note: you may need to register an account before downloading. If not, then the "Download" button is just pointing to the register page, and no actual file is downloaded.
When in doubt, do a web search. Make use of quotes (e.g., "vibrant" and "stock 2.2.1") in order to narrow the results and find what you need.
A collection of shipped ROMs. Contains ROMs for 40-odd devices.
Translated from a popular Chinese ROM, MIUI is a highly customized version of Android with a large base of customization and theme options and a strong iOS aesthetic. Available for many devices and in Froyo (2.2), Gingerbread (2.3), Ice Cream Sandwich (4.0), and Jelly Bean (4.3) options.
Android Open Kang Project (AOKP) is just like Cyanogen Mod (CM) a Android Open Source Project (AOSP) based ROM. AOKP focuses on providing extensible settings with the disadvantage that fewer devices are supported in comparison with CM.
Another popular Android forum, similar to XDA and RootzWiki.
Similar to CyanogenMod, OmniROM is a committed open-source project that provides ROMs for a variety of devices.
Similar to OmniROM, Replicant is a committed open-source project that provides ROMs for several devices (mostly Samsung). It is a fully free Android distribution (with a marked aversion to binary blobs), putting the emphasis on freedom and privacy/security.
Lineage OS is the successor to CyanogenMod (in the sense that the numbering scheme matches only). It allows for customization of the buttons at the bottom of the screen and for root access.
Flashtool provides an inbuilt tool - "XperiaFirm" that lets you download the stock firmware for Xperia devices. Using flashtool you can even install the downloaded firmware onto your Xperia device.
The website provides original & custom ROMs for more than 65 brands (mostly Chinese). In addition, it contains very useful Tools with their latest updates.
Contains a lot of MTK-based brands (mostly Chinese), and tools and drivers also are available there.
Similar to MTK Firmware.
Large list of Android ROMs.