Multiple access communication systems enable numerous users to share common communication resources, playing a crucial role in wireless networks. With the emergence of the sixth generation (6G) and beyond communication networks, supporting massive machine-type communications with sporadic activity patterns is expected to become a critical challenge. Unsourced random access (URA) has emerged as a promising paradigm to address this challenge by decoupling user identification from data transmission through the use of a common codebook. This survey offers a comprehensive overview of URA solutions, encompassing both theoretical foundations and practical applications. We present a systematic classification of URA solutions across three primary channel models: Gaussian multiple access channels (GMACs), single-antenna fading channels, and multiple-input multiple-output (MIMO) fading channels. For each category, we analyze and compare state-of-the-art solutions in terms of performance, complexity, and practical feasibility. Additionally, we discuss critical challenges such as interference management, computational complexity, and synchronization. The survey concludes with promising future research directions and potential methods to address existing limitations, providing a roadmap for researchers and practitioners in this rapidly evolving field.