The paper is mainly devoted to the robust stability problem of non-linear QFT designs. The problem is first formulated for SISO non-linear systems, limited in practice to linear-memoryless sector-bounded nan-linear interconnections, and in an I/O stability sense. The work investigates several possible robust adaptations of the Circle Criterion, depending on the type of resulting interconnection of linear and non-linear blocks, appearing in the feedback system. Also, the use of the Popov Criterion is investigated as an alternative less conservative than the Circle Criterion in some cases. The proposed techniques are given in usual QFT language, expressed as frequency conditions or boundaries in the Nichols Chart, allowing an easy integration with other design objectives. In addition, multivariable extensions using a conicity condition and the concept of maximal cone are adopted to give stability boundaries in interconnections of SIMO linear-MISO sector-bounded memoryless blocks. All the robust stability criteria are illustrated using significant examples to emphasize the practical application of the resulting techniques.