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Once you start using Office 365 you will probably notice that the interface for adding web parts is different than in SharePoint 2010. No worries though, because the changes are good ones! Below are some screenshots that highlight the process for adding web parts to pages in Office 365.
First up- access the page where you want to add the web part and then select to edit the page.
Now, enter your cursor in the location where you want to add the web part.
Select the Insert Ribbon. There will be a group called Web Parts that displays several of the common types of web parts that you can insert on the page.
Notice that each of these has a drop down arrow. Selecting this arrow will allow you to see the existing list of that type or create a new list of that type.
If you want to add a web part that isn't displayed on the main menu, then you can click on more web parts to see a list of all the web parts that can be added to the page. When you select the More Web Parts option a Silverlight application page will be displayed that allows you to select the web parts you want to add to the page. From this page you can either see all web parts, filter by category or search for specific web parts.