This survey provides an overview of the challenges of misspellings in natural language processing (NLP). While often unintentional, misspellings have become ubiquitous in digital communication, especially with the proliferation of Web 2.0, user-generated content, and informal text mediums such as social media, blogs, and forums. Even if humans can generally interpret misspelled text, NLP models frequently struggle to handle it: this causes a decline in performance in common tasks like text classification and machine translation. In this paper, we reconstruct a history of misspellings as a scientific problem. We then discuss the latest advancements to address the challenge of misspellings in NLP. Main strategies to mitigate the effect of misspellings include data augmentation, double step, character-order agnostic, and tuple-based methods, among others. This survey also examines dedicated data challenges and competitions to spur progress in the field. Critical safety and ethical concerns are also examined, for example, the voluntary use of misspellings to inject malicious messages and hate speech on social networks. Furthermore, the survey explores psycholinguistic perspectives on how humans process misspellings, potentially informing innovative computational techniques for text normalization and representation. Finally, the misspelling-related challenges and opportunities associated with modern large language models are also analyzed, including benchmarks, datasets, and performances of the most prominent language models against misspellings. This survey aims to be an exhaustive resource for researchers seeking to mitigate the impact of misspellings in the rapidly evolving landscape of NLP.