Time Travelling

I sometimes see myself travelling through time, changing some events that changed my life and history around the world, but then comes the question: would I still be me?
Some of the scars acquired through the years represent what we really are and how we defined our morale, knowledge and so on. How would changing those milestones would affect us?
Movies often take a peek at this theme, and TV series, as Heroes, also take their chance.
We have many examples of different behaviour of the characters as they are faced with the break of time-space continuum: some keep the memories of their original timeline, others are completely changed, as mind-erased copies of themselves; and yet others can keep a full record of all the new timelines created as they jump in the decades of their lives (eg. Butterfly Effect).
What is the right behaviour or the answer for questions around time-travelling, I still don't know; for those Terminator acquaintances around, maybe you know the answer.