Noncoherent communication is a promising paradigm for future wireless systems where acquiring accurate channel state information (CSI) is challenging or infeasible. It provides methods to bypass the need for explicit channel estimation in practical scenarios such as high-mobility networks, massive distributed antenna arrays, energy-constrained Internet-of-Things devices, and unstructured propagation environments. This survey provides a comprehensive overview of noncoherent communication strategies in multiple-input multiple-output (MIMO) systems, focusing on recent advances since the early 2000s. We classify noncoherent communication schemes into three main approaches where CSI-free signal recovery is based on subspace detection (i.e., Grassmannian signaling), differential detection, and energy detection, respectively. For each approach, we review the theoretical foundation and design methodologies. We also provide comparative insights into their suitability across different channel models and system constraints, highlighting application scenarios where noncoherent methods offer performance and scalability advantages over traditional coherent communication. Furthermore, we discuss practical considerations of noncoherent communication, including compatibility with orthogonal frequency division multiplexing (OFDM), resilience to hardware impairments, and scalability with the number of users. Finally, we provide an outlook on future challenges and research directions in designing robust and efficient noncoherent systems for next-generation wireless networks.