Let D be an integral domain and S be a multiplicative subset of D. Then given a semistar operation ? on D, we introduced the S-˜?-Noetherian domains, where ˜? is the stable semistar operation of finite type associated to ?. Among other things, we provide many different characterization for S-˜?- Noetherian domains by focusing on primary decomposition, weak Bourbaki associated primes and Zariski-Samuel associated primes of the S-saturation of a given quasi-˜?-ideal I of D.