Many feasibility and fixed-point problems in modern applications involve families of nonexpansive operators that do not necessarily possess a common fixed point. This talk presents two complementary approaches to such “inconsistent” settings. In the first part, I introduce a general dynamic string-averaging scheme in which the string structure, weights and control may vary across iterations. The method is perturbation-resilient, making it compatible with the superiorization methodology and it converges to a meaningful target even when the underlying operators share no fixed point. This yields a flexible and computationally efficient framework for tackling inconsistent feasibility problems. In the second part, I adopt a generic approach. For broad families of operators modeled on an underlying probability space, the set of exact common fixed points may be empty; nevertheless, under suitable conditions one can still establish the existence of generic almost common fixed points in the sense of Baire category—points fixed by all operators in a certain residual (comeager) subset, except possibly for a probabilitymeasure-zero subset. I will discuss generic existence results for such almost common fixed points; the abundance of solutions to these almost common fixed point problems is known to play a crucial role in many optimization algorithms. Together, these two directions offer a unified perspective on problems without guaranteed common fixed points, combining algorithmic convergence, perturbation resilience and superiorization with existence insights arising from genericity theory.