Does the French Expression 'Malgré Que' Take the Subjunctive?

Malgré que ("even though, despite the fact that") is a conjunctive phrase (locution conjonctive) that requires the subjunctive when there's uncertainty or supposition, as in:
Il le fait malgré qu'il pleuve.
He's doing it even though it's raining.

Je suis venu malgré que je n’aie pas vraiment le temps.
I came even though I really don’t have time.

The Heart of the Subjunctive

This goes to the heart of the subjunctive mood, which is used to express actions or ideas that are subjective or otherwise uncertain, such as will/wanting, emotion, doubt, possibility, necessity and judgment.

The subjunctive can seem overwhelming, but the thing to remember is: the subjunctive = subjectivity or unreality. Use this mood enough and it will become second nature...and quite expressive.

The French subjunctive is nearly always found in dependent clauses introduced by que or qui, and the subjects of the dependent and main clauses are usually different. For example:

•    Je veux que tu le fasses. > I want you to do it. 
•    Il faut que nous partions. >  It is necessary that we leave.

Dependent Clauses Take the Subjunctive When They:

1. Contain verbs and expressions that express someone's will, an order, a need, a piece of advice or a desire
2. Contain verbs and expressions of emotion or feeling, such as fear, happiness, anger, regret, surprise, or any other sentiments
3. Contain verbs and expressions of doubt, possibility, supposition and opinion
4. Contain verbs and expressions, such as croire que (to believe that), dire que (to say that), espérer que (to hope that), être certain que (to be certain that), il paraît que (it appears that), penser que (to think that), savoir que (to know that), trouver que (to find/think that) and vouloir dire que (to mean that), which only require the subjunctive when the clause is negative or interrogatory. They do not take the subjunctive when they are used in the affirmative, because they express facts that are considered certain—at least in the speaker's mind.
5. Contain French conjunctive phrases (locutions conjonctives), groups of two or more words that have the same function as a conjunction and imply supposition. 
6. Contain the negative pronouns ne ... personne or ne ... rien, or the indefinite pronouns quelqu'un or quelque chose.
7. Follow main clauses containing superlatives. Note that in such cases, the subjunctive is optional, depending on how concrete the speaker feels about what is being said. 

Why 'Malgré Que' Takes the Subjunctive

Malgré que is one of the conjunctive phrases (locutions conjonctives) described in number 5, many of which are listed below. These require the subjunctive because they imply uncertainty and subjectivity; it is best to try to memorize them, although you can also decide according to the meaning of the tense. Malgré que belongs to a subset of this category called opposition conjunctions, such as bien que, sauf que, sans que and others.

These Conjunctive Phrases Take the Subjunctive

• à condition que > provided that
• à moins que > unless
• à supposer que > assuming that
• afin que > so that
• avant que > before
• bien que > although
• de crainte que > for fear that
• de façon que > so that, in order that, in such a way that
• de manière que > so that
• de peur que > for fear that
• de sorte que > so that
• en admettant que > assuming that
• en attendant que > while, until
• encore que > even though
• jusqu'à ce que > until
• pour que > so that
• pourvu que > provided that
• quoique > even though
• quoi que > whatever, no matter what
• sans que > without

Additional Resources

The French Subjunctive
French Conjunctions
The Subjunctivator!
Quiz: Subjunctive or indicative?