This paper deals with the computation of finite stability regions for systems with a single nonlinear clement (tho Lur'o problem), and also for systems with a product-typo non-linearity. The Liapunov functions arise from the proofs of tho absolute stability theorems, such as the Popov criterion and the circle criteria. The new feature of this paper is the use of Liapunov surfaces which are not necessarily closed; it is shown that this leads to a considerable improvement of the estimate of the region of attraction of the null solution.