If you are using SharePoint Server 2010 or SharePoint Server 2013 on-premises in conjunction with Office 365 and SharePoint Online, then you probably have started to tire of having a search center for your on-premises content and a separate search center for your cloud-based content.
At Microsoft Ignite (Chicago, May 2015), it was announced that a Hybrid Search Application would be shipped that would build content search indexes for your on-premises data and ship them up to SharePoint Online so that you could rely on the one search center in SharePoint Online to be the search center source of truth for all of your content.
This was released a couple of months ago, in two ways. You can find it in the cumulative updates for SharePoint Server 2013 starting in September 2015, or in the Preview release of SharePoint Server 2016.
If you are running SharePoint 2010 search today, there were a few SharePoint 2010 search features that were replaced with different features in SharePoint 2013 search and others that may have been modified for SharePoint Online search. Please check the documentation in the Cloud hybrid search download for the list of unsupported search features.