the universe in matter terms has an end, the point at which no matter of any form is, is the end of the universe
Because 'Space' is a false vacuum we can also assume space has an end as well, a boundry of sorts, what would happen we we to cross this boundry I cannot say, I doubt the human conciousness is capable of such thought (it boggles the mind!

)it's possible you would cease to exist, maybe god lives there, perhaps you would exit one end of the universe and enter the other side. In my opinion however by crossing the boundary you would be taking a part of the universe with you (youself) thus extending the boundry and making it as though you never had crossed
As einstien said Time and Space are relative, therefore connected (hence you hear things like 'the space time continueum!') as a result measuring the unvierse or 'space' can probably only be done in terms of time, the universe is aproxamately 29 billion years across in every direction, (assuming the BBT sent space in equal speeds the universe would have a radius of 14.4 billion years)
therefore you would need to 'travel faster than time' for anywhere between 14.4billion years and 0 seconds (it's unknown how close to the centre of the universe we are, we could infact be right on the edge of the universe)
I personally Like the idea of the 'Void' from Doctor Who, that the edge of the universe is outside of time itself