【报告摘要】Let A be a (real or complex) ∗-algebra and let S be a family of ∗-representations of A on Hilbert spaces. Roughly speaking, a noncommutive Positivstellensatz expresses elements of A that are mapped into positive operators by all representations of S in terms of quadratic modules defined in algebraic terms. There are a number of different versions of noncommutative Positivstellensätze. The simplest one writes positive elements as a hermitean square a∗a, where a ∈ A. The next version expresses positive elements as a finite sum of hermitean squares. Another version allows ourselves to use denominators. A number of known results for each of these versions are reviewed. We give Positivstellensätze for the Weyl algebra and for the enveloping algebra of the ax + b group. The counter-part of the Stengle-Krivine Positivstellensatz for the Weyl algebra is stated as an open problem.