You need to check which configurations IntelliJ will use to provide inspections on wired beans.
Go to File➝Project Structure➝Project Settings➝Modules.
Select your module and selected (or add) the Spring facet.
In the right area you can add or modify application contexts that are used by your module. Just check every configuartion you need in your context.
In addition to the above steps (in Bart's answer) , I had to
1. click on the unmapped spring beans from the warning dialog box
2. select all the beans file
3. set application context as "Spring Application Context" (You many want to change it to other contexts, if you need)
I just wanted to add that for our project (idea 13), looking at File ➝ Project Structure ➝ Project Settings ➝ Modules (Spring) everything looked correct, but that then when you highlight all of them and click the edit pencil that is when we could truly see the ones that it is complaining about. In our case they were all in our exploded war directory, so they only showed up after a full initial build.
Same situation with my team, appeared that the "Import Maven project automatically" (Settings / Maven / Importing) is not checked on a fresh IntelliJ installation.
So check it and next time you'll import a Spring based project pom.xml file in "Maven projects" window, then a green window will appear up right and will wait for your validation to trigger the parsing of Spring configuration files detected.
Hope it helps!
EDIT : at this time, using JDK 1.7 instead of 1.8 + start from a extra-simple fresh "empty project" (IntelliJ) had seemed necessary too on a other colleague laptop
If you are working with Gradle in Intellij, try the refresh ("Reload All Gradle Projects") button. It should make all of your Spring and other annotations recognizable by IDE, assuming you have properly defined your dependencies in build.gradle file.
Many answers or 6+ years old, so this is more current 2021-06 and while they are all still accurate AFAIK, the UI now gives a link to a help page that I'm including next.