As we delve into a life governed by AI, ongoing research continues to discover new forms of intelligence that are efficient and closely mimic an organism's brain in terms of performance. This article presents a new concept termed wet-neuromorphic computing, where neural network or similar structures from living cells and organisms are extracted to do computing. The wet-neuromorphic computing concept allow the (i) use biological cells or organisms to perform computing task by exploiting their natural molecular functions as living machines, or (ii) get inspiration to create novel bio-inspired algorithms diverging from traditionally from neural systems. We first present background on neuromorphic computing, and match their key properties to biological AI neural networks that are found in bacteria, 3D organoids, as well as neural system of C.Elegans. In each of these systems, we analyze their natural computing functions and review state-of-the-art research that has proposed using each of these systems for AI computing. As a case study, we detail experiments using bacterial cells for AI computing, harnessing their gene regulatory mechanisms to control growth levels, where we present wet-lab experimental results demonstrating the use of bacteria for pattern recognition application. Lastly, we discuss challenges and future directions for wet-neuromorphic computing.