Even native speakers have no end of trouble with the distinction between aun and aún. In fact, this newsletter was instigated when our chief proofreader removed the accents she found on the u's in Belanova's title refrain Y aun así te vas. The band's own CD shows an accent on the u, so we were dubious. Ultimately, she convinced us that there should be no accent on the u in the phrase aun así. Hopefully the following will convince you too!

Y aun así te vas
"And even so you leave"
[Captions 17-9, 22-4, 29-31, 34 Belanova > Y aun así te vas]

Aun así is a Spanish idiom, or usage expression, meaning "even so", "still" or "yet." We could have equally well translated the line as "And still you leave," or "And yet you leave."

Hace frio afuera, aun así ella no se pone un abrigo.
"It's cold outside, yet she won't wear a coat."

No tengo mucho dinero, pero aun así voy a comprar la computadora.
"I don't have much money, but still I'm going to buy the computer."

Habíamos pagado por la habitación y aun así tuvimos que buscar otro hotel.
"We had paid for the room and yet we had to look for another hotel."

Aun así, creo que deberías disculparte.
"Even so, I think you should apologize."

The word aun, by itself, and with no accent over the u, and not followed by así, can often be translated as "even."

No como torta, aun en mi cumpleaños.
I don't eat cake, even for my birthday.

Aun cuando lo leyera, no lo entendería.
I wouldn't understand it, even if I read it.

Ni aun sabiendo la dirección llegarías a su casa.
Not even knowing the address would you find his house.

Can you see how when we put aun together with así ("like this" / "this way"), we get something along the lines of "even like this" / "even this way"? Or, more concisely, "even so"? Diccionario de Uso del Español, by María Moliner, a favorite of professional translators, goes deeper:

If that's a bit too deep, ¡no importa! (don't worry), just remember the basic meaning and you'll be fine!

Aún, with the accent on the ú, means "up until the present moment" and is basically synonymous with todavía. Confusingly enough, aún is also defined as "yet," "still," but in the temporal sense (as opposed to when they mean "even so" / aun así).

¿Aún estás aquí?
¿Todavía estás aquí?
Are you still here?

Aún no ha llamado.
Todavía no ha llamado.
She hasn’t called yet.

Ya son las once y aún no ha llamado.
Ya son las once y todavía no ha llamado.
It’s already eleven o’clock and she still hasn’t called.

¿Has tenido noticias? —Aún no
¿Has tenido noticias? —Todavía no
Have you had any news? — Not yet.