Direction-of-arrival (DOA) estimation plays a crucial role in array signal processing across various domains, including radar, sonar, wireless communications, and seismic exploration. However, traditional DOA techniques often assume either far-field (FF) or near-field (NF) propagation, limiting their applicability in scenarios involving mixed-field sources. DOA estimation and localization in scenarios involving mixed NF and FF sources is a complex and dynamic field that has garnered significant research attention in recent years. This multifaceted and evolving area holds promise for addressing challenges in radar, wireless communications, and acoustic sensing applications. This review paper provides a comprehensive overview of the methodologies, techniques, and advancements in this domain. We categorize existing methodologies, discussing their advantages and limitations. Furthermore, we delve into the mathematical modeling of mixed-field sources and essential signal processing techniques for parameter estimation. Special attention is given to technical issues such as aperture loss, computational complexity, and hardware considerations. The paper discusses the various sources of noise in the mentioned scenario and highlights the importance of modeling noise accurately for effective estimation. It also explores different scenarios and assumptions considered in the literature, ranging from non-Gaussian and nonstationary noise environments to scenarios involving multipath propagation and unknown mutual coupling effects. A detailed examination of the statistical approaches used in DOA estimation and localization reveals a diverse range of methods, including higher-order statistics and second-order statistics, each with its own advantages and applications. A comparative evaluation of various approaches highlights their performance in terms of estimation accuracy, resolution, aperture loss and computational efficiency. This provides insights into the trade-offs involved in choosing between different approaches. The review also identifies promising future research directions, such as the exploration of advanced signal processing techniques like compressive sensing and deep learning, exact NF modeling, estimation based on one-bit measurements, the integration of polarization diversity, employing metasurface antennas, tracking parameters, and the utilization of full-wave or experimental data for a more realistic representation of the challenges. By reviewing advances in methodologies and techniques, as well as outlining future research directions aimed at addressing the complexities of mixed-field scenarios, this paper paves the way for the development of more robust and reliable localization systems capable of handling real-world complexities.