As an immortal, you are in some way exempt from being tied to the normal mortal coil. You're either fully immortal, reincarnated immortal, or ageless but vulnerable. The latter type requires a lot of intelligence and careful planning if you don't want to be discovered. You should ideally have a lot of resources financially, and certain contacts, like people who can change your identity. Also it helps to be VERY mentally resilient, because We Are as Mayflies and so forth. There's an eternal life that does not include eternal youth, leaving you in worse shape every year but knowing that your condition has nowhere to go but down.
And then there's the whole "can be mutilated horribly or torn apart but not die from it" scenario, often the result of an uncareful wish from a Literal Genie, leading to A Fate Worse Than Death, often followed by And I Must Scream as you're reduced to a mangled pile of still-sentient ground chuck.
This Cracked article points out the human perspective on time as you get older, meaning that everything becomes inconsequential the older you get, including relationships. And even if you're fortunate to avoid all of the above, you will get cancer at some point down the road. If you are both immortal and invulnerable, you will survive events such as the eventual death of the sun and be floating in space, unlikely to ever see another object again, and never dying, not even of starvation, dehydration or exposure (but may well feel those things) - you're there until the heat death of the universe. An entire page can be written on this, truthfully.