Update 13/6/2017: There have been numerous changes in the way Google Photospheres are published and viewed in the past 12 months. For a long time the embed option wasn't available but Google have reinstated it and provided tools for setting the iFrame size and previewing the result. I've updated the post to highlight the changes but I've retained the old, now redundant method, just in case Google take away the embed option again.

Update 21/1/2016: Jim Popenoe just posted that Google have made some progress updating Google Maps and it's now possible to embed Photospheres without requiring Javascript or an API key. In 5 simple steps you can embed your Photospheres using an iFrame. Read more to see how.