In the age of social media and generative AI, the ability to automatically assess the credibility of online content has become increasingly critical, complementing traditional approaches to false information detection. Credibility assessment relies on aggregating diverse credibility signals - small units of information, such as content subjectivity, bias, or a presence of persuasion techniques - into a final credibility label/score. However, current research in automatic credibility assessment and credibility signals detection remains highly fragmented, with many signals studied in isolation and lacking integration. Notably, there is a scarcity of approaches that detect and aggregate multiple credibility signals simultaneously. These challenges are further exacerbated by the absence of a comprehensive and up-to-date overview of research works that connects these research efforts under a common framework and identifies shared trends, challenges, and open problems. In this survey, we address this gap by presenting a systematic and comprehensive literature review of 175 research papers, focusing on textual credibility signals within the field of Natural Language Processing (NLP), which undergoes a rapid transformation due to advancements in Large Language Models (LLMs). While positioning the NLP research into the the broader multidisciplinary landscape, we examine both automatic credibility assessment methods as well as the detection of nine categories of credibility signals. We provide an in-depth analysis of three key categories: 1) factuality, subjectivity and bias, 2) persuasion techniques and logical fallacies, and 3) check-worthy and fact-checked claims. In addition to summarising existing methods, datasets, and tools, we outline future research direction and emerging opportunities, with particular attention to evolving challenges posed by generative AI.