You can travel within the EU right up until the last minute of the last day (in June) specified in your passport. (That, of course, assumes that you hold a UK, or other EU, passport).
The same is true of travel to most, but not all, other countries. (A very small number of countries require you to have a specified amount of time left on your passport when you visit them. This is often three or six months).
If you renew your passport now, any unexpired time will be added to the validity of your new passport.
http://www.ips.gov.uk/passport/apply-renewing. asp
Chris