If you've ever had to move a website, you've probably been faced with the fear that Google might not figure out that the website is new and it will suffer in the search results until they manage to sort out what happened.
Google has updated their guidelines to help webmasters deal with moving sites, including issues that might come up due to moving a site to responsive design.
Google has also updated information to include moving a site with no URL change, such as if you were to go to a new hosting provider or a new platform that was able to keep the old URLs intact.
When it comes to moving a site that has completely new URLs, Google goes into quite a bit of detail on how webmasters should map old URLs to new ones, and things to also keep in mind to fix any stragglers that somehow get missed in the transition.
If you have a website that serves different URLs for mobile users versus desktop users, also remember that Google has new smartphone recommendations for webmasters that detail exactly how to handle these kinds of situations, as well as benefits of using responsive design to alleviate these issues.
When you move your site, you can use Google's change of address tool within Google Webmaster Tools. You need to have both sites verified, both the old domain and the new domain, and it is quite simple to do by following instructions.