They always contain an idea that somehow contradicts the idea of the principal clause.

Though it was late the streets were full of people.

They answer the question: in spite of what?

The Subjunctive mood can be used in adverbial clauses of concession introduced by the conjunctions and pronouns though, although, however, matter how, whatever, wherever, whenever, whoever, etc. The Analytical Subjunctive with the mood auxiliaries may or might is used here.

Какой бы интересной ни была эта проблема, мы не можем сейчас обсуждать ее.
However interesting this problem might be we can't discuss it now.

The word order is usually inverted here, - a predicative, expressed by an adjective or participle stands before the subject.

Adverbial clauses of concession can be also introduced by the conjunction as.

Difficult as the problem might be we'll have to solve it.
Как бы ни была трудна эта проблема, нам придется ее решать.

However badly he might have behaved in the past now he is ideal husband.
Как бы скверно он не вел себя в прошлом, сейчас он идеальный муж.

The Analytical Subjunctive with the mood auxiliaries may and might plus Perfect Infinitive can be used in adverbial clauses of concession when the speaker describes a situation in the past which is different from the situation at present.

Note: Bear in mind that Indicative mood can be also used in adverbial clauses of concession both in English and in Russian.

However dark the night is we'll have to continue our way.
Как ни темна ночь нам придется продолжать путь.

In this case the action in the subordinate clause is represented as a fact not as supposition.

Young as the King was, he was known a one of the braved Knights of Europe.
Как ни молод был король, он был известен как храбрейший рыцарь Европы.