ASP.NET Web API application gives 404 when deployed at IIS 7

For me, in addition to having runAllManagedModulesForAllRequests="true" I also had to edit the "path" attribute below. Previously my path attribute was "*." which means it only executed on url's containing a dot character. However, my application's url's don't contain a dot. When I switched path to "*" then it worked. Here's what I have now:

  <system.webServer>
      <validation validateIntegratedModeConfiguration="false" />
      <modules runAllManagedModulesForAllRequests="true">
      <remove name="WebDAVModule"/>
      </modules>

      <handlers>
          <remove name="WebDAV" />
          <remove name="ExtensionlessUrlHandler-ISAPI-4.0_32bit" />
          <remove name="ExtensionlessUrlHandler-ISAPI-4.0_64bit" />
          <remove name="ExtensionlessUrlHandler-Integrated-4.0" />
          <add name="ExtensionlessUrlHandler-ISAPI-4.0_32bit" path="*" verb="*" modules="IsapiModule" scriptProcessor="%windir%\Microsoft.NET\Framework\v4.0.30319\aspnet_isapi.dll" preCondition="classicMode,runtimeVersionv4.0,bitness32" responseBufferLimit="0" />
          <add name="ExtensionlessUrlHandler-ISAPI-4.0_64bit" path="*" verb="*" modules="IsapiModule" scriptProcessor="%windir%\Microsoft.NET\Framework64\v4.0.30319\aspnet_isapi.dll" preCondition="classicMode,runtimeVersionv4.0,bitness64" responseBufferLimit="0" />
          <add name="ExtensionlessUrlHandler-Integrated-4.0" path="*" verb="*" type="System.Web.Handlers.TransferRequestHandler" preCondition="integratedMode,runtimeVersionv4.0" />
      </handlers>
  </system.webServer>

While the marked answer gets it working, all you really need to add to the webconfig is:

    <handlers>
      <!-- Your other remove tags-->
      <remove name="UrlRoutingModule-4.0"/>
      <!-- Your other add tags-->
      <add name="UrlRoutingModule-4.0" path="*" verb="*" type="System.Web.Routing.UrlRoutingModule" preCondition=""/>
    </handlers>

Note that none of those have a particular order, though you want your removes before your adds.

The reason that we end up getting a 404 is because the Url Routing Module only kicks in for the root of the website in IIS. By adding the module to this application's config, we're having the module to run under this application's path (your subdirectory path), and the routing module kicks in.