With the fast-paced development of digital innovations, web technologies are on the rise. With this in mind, making the web accessible for all became an interesting line of research and development. Web accessibility means that websites, tools, and technologies are designed and developed so that people with disabilities can use them with minimal to no supervision. More specifically, people can perceive, understand, navigate, and interact with the web. Over the past years, there have been attempts to achieve web accessibility for web users with physical, visual, and sensory disabilities. However, people with cognitive disabilities were not seen as a user group, thus, leaving them far behind the target general population for improving web accessibility. This paper presents an overview of the current state of web accessibility for people with cognitive disabilities. This paper attempts to come up with a unified definition of web accessibility and identify its components. The study identified the common reasons for the non-conformity of web accessibility. Moreover, the different cognitive disabilities and their linked issues are presented. Furthermore, this study reviews the various interventions made by governments and private organizations worldwide. The study uncovered that web accessibility for people with cognitive disabilities hadn’t been a priority for research which resulted in non-conformities with the accessibility guidelines such as poor interface design, confusing guides and instructions, robustness, and lack of features to cater to the needs of people with cognitive disabilities. Results show that in each kind of cognitive disability, namely Deficit Hyperactivity Disorder (ADHD), Autism Spectrum Disorder (ASD), Intellectual Disability, Memory Impairments, Perceptual Disability, and Seizure Disorder, people experience various difficulties in accessing the web, specifically on processing too much content or information, confusing instructions, typographies, and grammar. Laws have been imposed on different Asian and Western countries to address these difficulties. In contrast, other countries and organizations have adopted the Web Accessibility Guidelines of the World Wide Web Consortium (W3C). In conclusion, as organizations gear toward web equality, governments have the initiative to move towards the goal by implementing different laws and policies driving the WCAG 2.0 as a basis of these to orchestrate the design of websites effectively. Furthermore, this study calls for action for extensive research on web accessibility for cognitive disabilities as it is not only valuable to the vulnerable sector of society but also for the rest of the population.