From the time man realised he could die it seems he has been fantasising about cheating death and continuing to live forever, from cavemen hoping to forever follow the great herds of animals to scientists trying to find that aging gene we have been obsessed to a greater or lesser degree of finding a way to ensure life doesn't end. Immortality is quite obviously the opposite of mortality, that stage of life associated with being dead. Therefore some one who is immortal will not die and living forever is a possibility.

"I'm Immortal! I can't Die!"

Well not quite. It is still possible to die if you are theoretically immortal; Hercules supposedly managed it for example¹, but that was down to his father being a God. It is usually accepted that humans that are immortal can be killed, but if they somehow manage to avoid that fate, can live forever.

Living Forever

There are a few ideas on this one depending on your culture, your beliefs and various stories you quite like the sound of.

• Becoming a God

  
  The original and some might say the best way of becoming immortal is to be a God. Many have tried but it can be a bit difficult convincing your friends as they will often try to kill you to prove the validity (or not) of your claim. A particular favourite of the living god kings of various ancient civilisations. Slightly harder to manage in a generally monotheistic or atheistic world.
  

• Deities as Your Parents

  
  Favoured by those in Greek myths but rather difficult to engineer for yourself. Having an all powerful being as a parent tends to give you an advantage, the shortcut to eternal life hopefully just one of the things in your genes. It would want to be a benefit though, Mummy or Daddy are unlikely to croak so you're never going to get your hands on the material inheritance.
  

• Selling your soul

  
  An option since the creation in some religions, you can always sell your soul to some infernal, but immensely powerful, beast in exchange for an extended life. This was the premise behind the story of "Faust" by Goethe. The negative side of this is that were you to die, you have the problem of spending the rest of eternity in eternal damnation or some version of hell. You also lack soul which may be particularly problematic if you ever wish to have a Motown recording contract.
  

• Games

  
  Traditionally it has been suggested that you can live forever by literally playing a game against Death² and winning. However he is also acknowledged as the foremost expert on just about every game going so you are unlikely to win.
  

• Wishes

  
  Just about everyone has tried this one in some form or another, whether making requests off a genie from a bottle á la Aladdin, or before blowing out candles on a Birthday cake. Not extremely effective historically but atttempting it doesn't cost you anything.
  

• Dying

  
  Dying however costs your life. Funnily enough though this is seen as a method of living forever. The major religions have to offer some reward for sticking by the rules and this seems to be it. Once you die your soul³ can live forever in some paradise. This is another that has yet to be proven but it offers possibilities if you don't manage the living forever thing in this life.
  

• Reincarnation

  
  Talking of other lives, reincarnation allows the possibility of living them, provided you die first and lose all knowledge of previous lives. At least you get to spend most of eternity in the same world as everyone else.
  

• Fame

  
  A more modern day version of immortality. Some humans have realised that there is no way that they can actually live forever, so instead have opted for fame hoping that they will live on in the minds of their fans and the rest of the general public. Unfortunately the majority of famous people leave the collective consciousness as soon as they leave the public eye, whether through becoming a recluse or popping their clogs.
  

• Science

  
  The final resting point of most people's hopes for conquering mortality, science offers a plethora of ways to beat off death, from the ultra-modern genetic engineering, to more simple medical improvements allowing extended lives through surgical procedures or by eliminating viruses. There is also the increasingly popular method of having your body(or parts thereof) frozen in liquid nitrogen for reanimation at a later date when science has progressed further.
  

Despite the huge number of ways to live forever, no-one has yet convincingly managed it⁴, thus indicating they may not be quite as foolproof as some adherents might think. Immortality may well be beyond us but that won't stop some people still looking.