The verb form that is usually preceded by να is traditionally called “subjunctive” (Υποτακτική) and can be used for a number of different cases. In general, it is used to present an event as wanted, expected, intended etc.

Η Κλειώ πρέπει να διαβάσει αλλά θέλει να πάει σινεμά

Formation
In the imperfective subjunctive the same forms that appear in the present of either voice follow the particle να.

να + present tense να αγαπάω να αγαπιέμαι

Perfective subjunctive forms consist of να and verb forms having no independent status, i.e. they are used only together with να (or ας). These forms are based on the perfective verb stem of either voice (αγαπησ- / αγαπηθ-).

να αγαπήσω/ να αγαπηθώ

Perfect subjunctive forms consist of the particle να and the same forms that appear in present perfect of either voice. να+present perfect tense να έχω αγαπήσει active

να έχω αγαπηθεί medio-passive

The negative forms take the particle μην (instead of δεν) placed just after the particle να and before the verb form.

Depending on what we want to emphasize, we can use either the perfective or the imperfective aspect.
However, there are certain verbs and phrases which must take either perfective or imperfective aspect. να μην μιλάω / να μην κοιμηθώ
Use

When preceded by να this verb form can be found a. used independently, in main clauses It can be used to express

• wish
• order or request
In this case, the subjunctive is used instead of the imperative, either because there is no imperative form (e.g. for passive imperfective) or because we want to make the order sound milder and politer. It can also be used to give an order in the third person.
• ask for permission
• give permission

b. used dependently, in complement clauses. It is often used after verbs which express desire, wish, intention, ability, possibility, probability etc.: θέλω / εύχομαι / ελπίζω / σκοπεύω / σκέφτομαι.The subjunctive verb form can also be preceded by the particle ας instead of να, but in this case it is only used in main clauses (see a. above).

There are cases where this verb form can be found alone, that is, without να or ας before it. This can happen a. in prohibitions (negative orders).
b. after πριν and ίσως
c. after όποιος / όποτε / όπου / όπως

You can also read: How to express wish

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