Despite and even though (a pesar de, a pesar de que)

"Despite": a pesar de + Noun or Infinitive

To say that something happens despite a physical obstacle or an opposing fact, use a pesar de (despite, in spite of). When using this phrase, it must be followed by a noun or an infinitive verb—never a conjugated verb.

Despite the tiredness, she does not give up.
A pesar del cansancio, ella no se rinde.
The doctor works despite being sick.
El médico trabaja a pesar de estar enfermo.

You can place the a pesar de phrase at the beginning or the end of the sentence. If it starts the sentence, use a comma to separate it from the main action.

Despite the warning, the boy insists on playing.
A pesar de la advertencia, el niño insiste en jugar.

"Even though": a pesar de que + Conjugated Verb

If you need to use a conjugated verb to express "even though" or "despite the fact that", add que to the phrase to make a pesar de que.

even though, despite the fact that
a pesar de que

When declaring a known, established fact or an action that is definitely happening, use the Indicative mood (your standard tenses) after a pesar de que.

My mother works even though she is very tired.
Mi madre trabaja a pesar de que está muy cansada.
Even though it is very difficult, we do not give up.
A pesar de que es muy difícil, no nos rendimos.

However, if the obstacle is hypothetical, a future possibility, or something the speaker is acknowledging without declaring it as a concrete fact (similar to saying "even if"), you must use the Subjunctive mood.

Even though the risk may be enormous, they insist on going.
A pesar de que el riesgo sea enorme, ellos insisten en ir.