In short

The infinitive, the participle and the gerund are Italian’s three non-finite moods (modi indefiniti): verb forms that never mark a person. Their job is to compress a whole subordinate clause into a short phrase with no conjunction - Dopo aver mangiato instead of dopo che ho mangiato, camminando instead of mentre camminavo. As a rule the compressed (implicit) clause borrows the same subject as the main clause.

The rule

They are called indefiniti because, unlike the indicative or subjunctive, they do not tell you who does the action - there is no io/tu/lui ending. That missing subject is supplied by the main clause. This is what lets them build a frase implicita: a subordinate clause whose verb is non-finite and whose subject is usually left unexpressed.

Each of the three moods has a present form and a past (compound) form. The past form always signals anteriority - the action happened before the main verb.

• Step 1 - The infinitive (l’infinito). Beyond naming the verb, the infinitive works two ways. (a) As a noun: it can be the subject or object of a sentence, sometimes with an article - Fumare fa male, Non dimenticare il passato è un dovere. Where English uses an “-ing” noun, Italian uses the infinitive. (b) As a past infinitive (aver / essere + participio) after a temporal preposition, above all dopo: Dopo averlo visto, gli ho parlato (after seeing him…). Same subject as the main clause.
• Step 2 - The gerund (il gerundio). The present gerund (-ando / -endo) is invariable and adverbial: it adds a time (while), means (by), manner or cause value to the main verb - L’ho incontrato camminando per strada, Sbagliando si impara, Essendo stanco, sono rimasto a casa. The past gerund (avendo / essendo + participio) marks anteriority: Avendo finito il lavoro, posso andarmene (having finished…). Its subject is normally the same as the main clause.
• Step 3 - The participle (il participio). The present participle (-ante / -ente) is now mostly frozen as adjectives and nouns - interessante, seguente, cantante, insegnante - and only rarely a live verb form; it agrees in number but not gender. The past participle (-ato / -uto / -ito) does more than build compound tenses: on its own it forms an absolute clause expressing anteriority - Fatta la spesa, sono tornato a casa. Here the participle agrees with its noun, and this construction may even carry its own subject: Partito il treno, siamo rimasti soli.
• Step 4 - Implicit <-> explicit. Every implicit clause can be unfolded into an explicit one (conjunction + finite verb): Dopo aver mangiato = dopo che ho mangiato; Essendo tardi = siccome era tardi. The catch is the same-subject constraint: you may compress to the implicit form only when the subordinate subject equals the main one. If the two subjects differ, keep the explicit form - or, with the participle/gerund, use the absolute construction that supplies its own subject.

Forms & patterns

The three moods, present and past:

Gerund endings - one for -are, one shared by -ere / -ire:

A handful of gerunds are built on the old Latin stem: fare -> facendo, dire -> dicendo, bere -> bevendo, porre -> ponendo, tradurre -> traducendo.

The implicit clause and its explicit twin:

The present participle, alive today mainly as ready-made adjectives and nouns:

Examples

Common mistakes

• Dopo aver finito, la mamma mi ha chiamato -> Dopo che ho finito, la mamma mi ha chiamato - the implicit dopo + infinito takes its subject from the main clause, so this reads “after Mum finished”; to say “after I finished” you must switch to the explicit dopo che + verbo.
• Fatto la spesa, sono tornato a casa -> Fatta la spesa, sono tornato a casa - in an absolute past-participle clause the participle agrees with its noun (la spesa, feminine), so it is fatta, not invariable.
• Mi piace nuotando -> Mi piace nuotare - an English “-ing” used as a noun (swimming) is the Italian infinitive; the gerund can never be a subject or object.
• un libro interessando -> un libro interessante - to modify a noun you need the present participle (-ante / -ente), not the gerund (-ando / -endo), which is adverbial.
• Dopo aver arrivata a casa -> Dopo essere arrivata a casa - verbs of movement take essere in the past infinitive, and the participle then agrees with the subject: arrivata for a woman.

Tips

• Non-finite = no person. Because these forms never say who, the subject is silently copied from the main clause - which is exactly why the same-subject rule is the default. When you catch yourself needing a different doer, reach for the explicit clause instead.
• English “-ing” splits in two. As a naming word it is the infinitive (eating = mangiare); as a “while / by doing” adverb it is the gerund (while eating = mangiando). Never the same Italian word.
• dopo loves the past infinitive. Read dopo aver / essere + participio as “after having done”, and mentally expand it to dopo che… to double-check the subject really matches.
• The present participle is vocabulary, not a factory. You rarely coin new ones; learn cantante, insegnante, interessante, seguente, corrente as words. For an ongoing action use stare + gerundio from step 16, not a participle.