Let \((X,\tau)\) be a Noetherian topological space (every strictly ascending chain of open sets is finite). The authors show that \((X,\tau)\) is sequentially compact and \(s\)-compact (related notions of semicompactness are discussed). Further, they define \(h(x)\), the height of \(X\) (related notions of dimension are discussed) and show that, if every irreducible closed subset of \(X\) has a generic point, then \(X\) is sequential if and only if \(h(X)\leq\omega_1\); this implies that the prime spectrum of a commutative Noetherian ring is a sequential Noetherian topological space.