The Office 2013 June 2013 Cumulative Updates are now available, please see the links below:
http://support.microsoft.com/kb/2855356
Project Server 2013 Server Roll up package June 2013 CU (Recommended):
http://support.microsoft.com/kb/2817415 (Delayed)
Project Server 2013 June 2013 CU (Included in the Server Roll up package):
http://support.microsoft.com/kb/2817341
Project 2013 June 2013 CU:
http://support.microsoft.com/kb/2817338
Also worth noting, install the March 2013 Public update: http://support.microsoft.com/kb/2768001 if installing the June 2013 CU.
The Office 2010 June 2013 Cumulative Updates are now available, please see the links below:
http://support.microsoft.com/kb/2855357
Project Server 2010 Server Roll up package June 2013 CU (Recommended):
http://support.microsoft.com/kb/2817368
Project Server 2010 June 2013 CU (Included in the Server Roll up package):
http://support.microsoft.com/kb/2817378 & http://support.microsoft.com/kb/2817387
Project 2010 June 2013 CU:
http://support.microsoft.com/kb/2817385
Remember SP1 is a pre-requisite for the Office 2010 June 2013 CUs.
For more details please see:
http://blogs.technet.com/b/projectsupport/archive/2013/06/14/microsoft-project-server-2007-2010-and-2013-june-2013-cu-announcement.aspx
As always, test these updates on a replica test environment before deploying to production
Modern Work Management - Project and the Power Platform blog 
Hi Paul, we haven’t updated since the August 2012 CU (missed the April 2013 ones). Is it ok to jump straight to the June 2013 CU for the server and the clients? I’m guessing it is as that’s the whole point of a CU isn’t it! thanks, Nick
Hi Nick, The CU’s are cumulative so you can “skip” patches. Which ever patch you decide to deploy, please fully test this on a replica Test farm before deploying to Production. Thanks Paul