It's the standard civil and criminal court at the city or provincial level. For immigration matters, dedicated sections of the tribunale handle asylum-related appeals and other immigration disputes.
If your asylum claim is rejected and you appeal, the tribunale's specialised section is where the appeal is heard. Deadlines are tight and a lawyer is usually needed.
Administrative cases (like visa or permit refusals) usually go to the TAR, not the tribunale ordinario. Asylum-related decisions go to the tribunale ordinario's dedicated section.
Official source
Ministero della Giustizia
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