In short

• Neuter nouns are the easiest gender to decline, because nominative, accusative and vocative are identical - in the singular and in the plural. Only the genitive is a separate form.
• Four everyday classes: -ο (το βιβλίο), -ι (το παιδί, το σπίτι), -μα (το μάθημα), and neuter -ος (το λάθος).
• Nouns in -ι build their genitives on -ιού / -ιών: του παιδιού, των σπιτιών.
• Nouns in -μα grow a -ματ- piece in every cell except the nominative/accusative/vocative singular: το μάθημα, but του μαθήματος, τα μαθήματα, των μαθημάτων. Deverbal nouns in -σιμο behave the same way: το γράψιμο, του γραψίματος.
• Neuter -ος is not masculine: το λάθος, του λάθους, τα λάθη, των λαθών.
• Foreign loans such as το γκαράζ, το ασανσέρ, το ταξί, το μπαρ, το πάρτι never change at all - the article alone shows case and number: τα ταξί.

The rule

A Greek noun is stored in your memory with its article, and for neuters that article is το in the singular and τα in the plural. That pair already does a lot of the work: because the neuter nominative, accusative and vocative are the same word, you only ever learn two shapes per number - the plain form and the genitive.

Which genitive you get depends on the ending you see in the dictionary form.

-ο. The commonest neuter type. The singular genitive is -ου, the plural is -α / -ων: το βιβλίο, του βιβλίου, τα βιβλία, των βιβλίων. Nothing is inserted; only the ending is swapped. A large group of (mostly learned) words stressed on the third syllable from the end moves the accent one step to the right in both genitives, because the endings -ου and -ων draw it onto the second-from-last syllable: το αυτοκίνητο → του αυτοκινήτου, των αυτοκινήτων, exactly like το πρόσωπο → του προσώπου, των προσώπων. That is a property of this group, not of the three-syllable rule (which would happily allow του αυτοκίνητου): everyday words of the very same shape keep their stress where it is, as in το σύννεφο → του σύννεφου, τα σύννεφα, των σύννεφων. Each word belongs to one group or the other and you learn it with the word - αυτοκίνητο shifts, σύννεφο does not. Only a small third group genuinely has both genitives, as in το βούτυρο → του βούτυρου / του βουτύρου.

-ι. Here the ending -ι is replaced by a syllable pronounced as one sound: -ιού in the genitive singular, -ιά / -ιών in the plural of oxytone words, -ια / -ιών in the plural of words stressed earlier. So το παιδί → του παιδιού, τα παιδιά, των παιδιών; but το σπίτι → του σπιτιού, τα σπίτια, των σπιτιών. Notice the asymmetry in σπίτι: the plural nominative keeps the original stress (τα σπίτια), while both genitives pull it onto the ending (σπιτιού, σπιτιών).

-μα. These nouns (almost all of them abstract or the result of an action) have a longer stem that only surfaces outside the nominative/accusative/vocative singular. The stem is the word minus -α, plus -ατ-: μάθημ- → μαθήματ-. Every remaining cell is built on it: του μαθήματος, τα μαθήματα, των μαθημάτων. In a word like μάθημα, stressed on the third syllable from the end, the accent moves forward twice, for two different reasons: in the genitive singular the extra syllable would leave it further back than the third-from-last (μάθημα → μαθήματος), and in the genitive plural the ending -ων pulls it one more step, onto -μά- (των μαθημάτων). Shorter ones stressed on the next-to-last syllable - το γράμμα, το χρώμα, το αίμα, το βήμα - have nowhere to move in the genitive singular (του γράμματος, του χρώματος) and shift only in the genitive plural (των γραμμάτων, των χρωμάτων).

-σιμο. Nouns formed from verbs - το γράψιμο “writing, handwriting”, το ντύσιμο “getting dressed”, το πλύσιμο “washing” - look like -ο words but decline like -μα words, with -ματ-: του γραψίματος, τα γραψίματα, των γραψιμάτων.

Neuter -ος. A small but very frequent group: το λάθος, το μέρος, το δάσος, το μέγεθος, το ύψος. They end in -ος like masculines, but they are neuter and their endings are completely different: genitive singular -ους, plural -η, genitive plural -ών with the accent on the ending. So το λάθος, του λάθους, τα λάθη, των λαθών. The two-syllable ones keep their stress where it is (λάθος, μέρος, δάσος, ύψος); a longer one stressed on the third syllable from the end moves it forward like any other word: το μέγεθος → του μεγέθους, τα μεγέθη.

Indeclinable loans. Words taken from French, English or Italian that do not fit any Greek pattern simply refuse to inflect: το γκαράζ, το ασανσέρ, το ταξί, το μπαρ, το πάρτι, το σινεμά, το μετρό. They are neuter by default, and the only thing that changes around them is the article and any adjective: το κίτρινο ταξί → τα κίτρινα ταξί. The trap is το ταξί, which ends in -ί exactly like το παιδί but is a loan, so its plural is τα ταξί, never *τα ταξιά.

Irregular neuters (το γάλα, το κρέας, το φως, το γεγονός) and learned classes are treated later in the path.

Forms & patterns

Neuter in -ο: το βιβλίο

Neuter in -ο with stress shift: το αυτοκίνητο

Neuter in -ι, stressed on the ending: το παιδί

Neuter in -ι, stressed earlier: το σπίτι

Same pattern: το τραπέζι, του τραπεζιού, τα τραπέζια, των τραπεζιών · το κλειδί, του κλειδιού, τα κλειδιά, των κλειδιών.

Neuter in -μα: το μάθημα

Deverbal neuter in -σιμο: το γράψιμο

Same pattern: το ντύσιμο, του ντυσίματος, τα ντυσίματα · το πλύσιμο, του πλυσίματος, τα πλυσίματα.

Neuter in -ος: το λάθος

Indeclinable loan: το ταξί

The same for every loan of this kind - only the article moves:

Summary of the endings

Examples

Common mistakes

• τα λάθοι -> τα λάθη - a noun in -ος is not automatically masculine; neuter -ος makes its plural in -η and its genitive plural in -ών (των λαθών, not *των λάθων).
• του λάθου -> του λάθους - the genitive singular of neuter -ος is -ους, never the masculine -ου.
• του μάθηματος -> του μαθήματος - adding -τος makes the word one syllable longer, so the accent must move forward to stay within the last three syllables.
• των μαθήματων -> των μαθημάτων - in the genitive plural the ending -ων pulls the accent one syllable further right than in the genitive singular, so it lands on -μά-.
• τα ταξιά -> τα ταξί - ταξί looks like παιδί but is an indeclinable loan; compare native το παιδί → τα παιδιά.
• στα γκαράζια -> στα γκαράζ - loans take no Greek ending at all; the article τα carries the plural.
• του παιδίου -> του παιδιού - nouns in -ι form the genitive in -ιού, plural -ιών.
• τα σπιτιά -> τα σπίτια - σπίτι keeps its original stress in the nominative/accusative plural; only the genitives (σπιτιού, σπιτιών) stress the ending.
• το μάθημα μας -> το μάθημά μας - a word stressed on the third-from-last syllable takes a second accent when a weak pronoun follows.

Tips

• Learn every neuter as a pair: plain form + genitive, singular and plural (το σπίτι - του σπιτιού · τα σπίτια - των σπιτιών). Because nominative, accusative and vocative coincide, those four forms are the whole declension.
• Use the ending as a decision tree: -μα and -σιμο → insert -ματ-; -ος → -ους / -η / -ών; -ι → -ιού / -ιών; -ο → -ου / -α / -ων.
• Whenever you build a genitive, check the accent before you write it: μάθημα → μαθήματος → μαθημάτων, αυτοκίνητο → αυτοκινήτου. See The three-syllable rule and stress shift.
• If a word sounds foreign, ends in a consonant (γκαράζ, ασανσέρ, μπαρ) or in a stressed vowel (μετρό, σινεμά), treat it as neuter and indeclinable, and let the article do all the grammar.
• Adjectives still inflect around an indeclinable noun: ένα μεγάλο πάρτι → δύο μεγάλα πάρτι.
• Neuter genitive plurals in -ιών, -μάτων, -ών are common in speech (των παιδιών, των μαθημάτων, των λαθών); do not avoid them the way you may avoid some feminine genitive plurals.
• το πρόβλημα, το σύστημα, το πρόγραμμα and all their -μα relatives are the same paradigm - learn μάθημα once and you have hundreds of words.