The exponential growth of textual and audiovisual information has made the task of automatic keyphrase extraction (KE) increasingly significant. This article provides a comprehensive analysis of contemporary theoretical approaches and practical solutions for KE across both text and speech modalities. The primary contribution of this work is its systematic synthesis of these often-disparate research strands into a unified analytical framework, highlighting the evolution of the field from statistical methods towards large language models (LLMs) and end-to-end speech processing. We examine the stages of KE, the characteristics of keyphrases in written and spoken language, and terminological nuances. Various methods for automatic KE are discussed and analyzed in detail: statistical, hybrid, machine learning-based, and structural. The review dedicates substantial attention to emerging paradigms, including keyphrase generation using LLMs, and provides a detailed overview of methodologies and challenges in automatic corpus annotation. Furthermore, we specifically analyze current directions and inherent difficulties in KE for spoken language, comparing transcript-based and end-to-end acoustic approaches. This synthesis leads us to conclude that the field is moving towards a more integrated, context-aware paradigm. Future progress will depend on addressing key challenges such as data scarcity for low-resource languages, effective multimodal fusion, and the nuanced evaluation of generative KE systems.