Flexible neuromorphic architectures that emulate biological cognitive systems hold great promise for smart wearable electronics. To realize neuro-inspired sensing and computing electronics, artificial sensory neurons that detect and process external stimuli must be integrated with central nervous systems capable of parallel computation. In near-sensor computing, synaptic devices, and sensors are used to emulate sensory neurons and receptors, respectively. In contrast, in in-sensor computing, a single multifunctional device serves as both the receptor and neuron. Bio-inspired cognitive systems efficiently detect and process stimuli through data structuring techniques, significantly reducing data volume and enabling the extension of neuromorphic applications to smart wearable systems. To construct wearable near- and in-sensor computing, it is crucial to develop artificial sensory neurons and central nervous synapses that replicate the biological functionalities. Additionally, the integrated systems must exhibit high mechanical flexibility and integration density. This review addresses research on flexible bio-inspired cognitive systems, classified into near- and in-sensor computing. It covers fundamental aspects, including biological cognitive processes, the required components, and the structures for each component, as well as applications for wearable smart systems. Finally, it offers perspectives on future research directions for flexible neuromorphic electronics in smart wearable systems connected to the next-generation Internet of Things.