The four pillars: cacio e pepe, carbonara, amatriciana, gricia. All use pecorino romano, not parmesan.
Carbonara: eggs + guanciale (pork cheek) + pecorino + pepper. No cream. Ever.
Cacio e pepe: pecorino + pepper + pasta water. Simple, requires technique. Try Roscioli or Felice a Testaccio for the best.