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Theme inheritance is very simple if you are working with publishing sites or sites where you manually activated the publishing features on site collection level. This feature depends up on version of SharePoint you are using and publishing feature is enabled or not
How to inherit sub-site theme / composed looks inheritance?
_Maybe the idea behind it is that you will only select a theme or composed look on the top site, and then inherit the sub-sites from the root sites’ one (I could be wrong, if so, ignore this sentence). _
How does composed looks inheritance work in SharePoint?
In the new version of SharePoint there is a significant difference between the previous one. The 2013 one works with a composed looks list (more information can be found here ). When you create your own Composed Look Items, these items are not inherited along the sub-sites.
In SharePoint 2013 this functionality is under master page settings page (Site Settings > Master Page) under the Theme section of that page. Please remember to mark your question as "answered" if this solves your problem.
One of the section background options. Cogwheel menu hover background in site contents view. Some button and checkbox backgrounds, links, texts, borders and icons, icon and link hovers. Also the footer background color, and one of the section background options. System pages: OK button background, link texts.
How do I inherit the CSS theme color in my JavaScript?
This way your app web will automatically take the styling of your host web. I just wrote a blog entry on how to do this on the server-side. Put the class ThemeHelper into your apps, and then generate your CSS on the server-side, placing theme colors into the CSS where necessary.
Instead of using fixed colors, SharePoint Framework allows you to refer to the theme colors of the context site. As a result, if your web part is placed on a site that uses a red theme, it uses the red palette as well, and if it’s placed on a site that uses the blue theme, it automatically adjusts itself to use the blue palette.