The gut microbiome has attracted increasing attention from researchers in recent years. The microbiota can have a specific and complex cross-talk with the host, particularly with the central nervous system (CNS), creating the so-called “gut–brain axis”. Communication between the gut, intestinal microbiota, and the brain involves the secretion of various metabolites such as short-chain fatty acids (SCFAs), structural components of bacteria, and signaling molecules. Moreover, an imbalance in the gut microbiota composition modulates the immune system and function of tissue barriers such as the blood–brain barrier (BBB). Therefore, the aim of this literature review is to describe how the gut–brain interplay may contribute to the development of various neurological disorders, combining the fields of gastroenterology and neuroscience. We present recent findings concerning the effect of the altered microbiota on neurodegeneration and neuroinflammation, including Alzheimer’s and Parkinson’s diseases, as well as multiple sclerosis. Moreover, the impact of the pathological shift in the microbiome on selected neuropsychological disorders, i.e., major depressive disorders (MDD) and autism spectrum disorder (ASD), is also discussed. Future research on the effect of balanced gut microbiota composition on the gut–brain axis would help to identify new potential opportunities for therapeutic interventions in the presented diseases.