The subjunctive is used after verbs expressing hope, wish, desire, command or doubt such as sperare - to hope, desiderare - to desire, volere - to want, and dubitare - to doubt.  But verbs that express certainty or fact used in the affirmative sense (and not negative) require the indicative, such as essere sicuro - to be sure, essere certo - to be certain, and sapere - to know.  And if the subject of both verbs in the sentence is the same, use di with the infinitive instead of the subjunctive.

Dubito che loro vengano.  I doubt that they'll come.
Spero che lei vinca.  I hope that she wins.
Non so se i musei siano aperti.  I don't know if the museums are open.
So che i musei sono aperti.  I know that the museums are open.
Non credo di averlo perso.  I don't think that I lost it.

The subjunctive is also used after impersonal expressions, usually essere and an adjective or adverb, unless they state a fact.  Some common expressions are è necessario - it's necessary, è meglio - it's better, è possibile - it's possible, and è probabile - it's probable.  The indicative is used after these expressions of certainty: è certo - it's certain, è sicuro - it's sure, and è vero - it's true.

Certain conjunctions require the subjunctive as well, such as sebbene - even though, benché - although, affinché - so that, prima che - before, purché - provided that, as long as and nel caso che - in the event that.  The subjunctive is also used after a relative superlative che, and after il primo.. che, l'ultimo..che, and il solo...che.