It looks like the tools for developing SharePoint 2013 customizations are mostly in Visual Studio 2012 and "Napa for Office 365."
If you've tried developing solutions for SharePoint 2013 in Visual Studio 2010, what has worked for you?
To add to Hugh's answer, almost anything you can do in VS 2010 for SP10 works in SP13 (event receivers, web parts, etc.), but nothing new, or not much of the new features work in VS2010 because much of them are new templates and data techniques such as the app model, remote event receivers, etc.
The App model requires the ability to do remote development which is built into VS 2012, but not 2010 and the SharePoint add on for VS is for 2012, not 2010.
By the way, it is Visual Studio 2012, not 2013, confusing, I know.
A positive push for VS 2012 is that remote development and deployment is allow, which matches Microsoft's push towards SharePoint online (office 365).
It seems to work for the most part, but I am not confident about things like workflows as I have had two not work now which work on 2010.
We are in the middle of upgrading visual studio now though, as a company we don't believe the risks of incompatibility outweigh the costs.
You can actually do loads still with VS2010 in SharePoint 2013. We do exactly that for DocRead (no time to upgrade to 2012). Check my post on supporting multiple versions of SharePoint from 1 VS Solution.
I just made an update to this article as I forgot to compile against .Net 4.0
Some relevant article you can find here: http://faisalrafique.wordpress.com/2012/12/09/whats-new-for-sharepoint-development-in-visual-studio-2012/