In-memory and in-sensor reservoir computing (RC) using memristive devices is a promising approach to address the energy and speed limitations of traditional digital computers in brain-like computing. This review discusses recent advancements in in-memory and in-sensor RC, including algorithm designs, material and device development, and applications in classification and regression. RC leverages the dynamic properties of memristive devices to enable real-time edge learning with reduced training complexity. The paper surveys three major RC models: echo state networks (ESNs), liquid state machines (LSMs), and single dynamic nodes with delayed feedback. It also introduces two representative RC architectures: in-memory and in-sensor, and discusses emerging materials and devices for these systems. Applications of RC include classification and regression tasks across various input modalities, such as images, audio, events, graphs, sequences, and multimodal data. The paper also discusses future perspectives on RC systems, including materials, architecture, and applications. In-memory and in-sensor RC systems offer significant energy efficiency and reduced training costs compared to traditional deep neural networks (DNNs). The review highlights the potential of memristive devices in enabling efficient and fast learning for intelligent edge systems.In-memory and in-sensor reservoir computing (RC) using memristive devices is a promising approach to address the energy and speed limitations of traditional digital computers in brain-like computing. This review discusses recent advancements in in-memory and in-sensor RC, including algorithm designs, material and device development, and applications in classification and regression. RC leverages the dynamic properties of memristive devices to enable real-time edge learning with reduced training complexity. The paper surveys three major RC models: echo state networks (ESNs), liquid state machines (LSMs), and single dynamic nodes with delayed feedback. It also introduces two representative RC architectures: in-memory and in-sensor, and discusses emerging materials and devices for these systems. Applications of RC include classification and regression tasks across various input modalities, such as images, audio, events, graphs, sequences, and multimodal data. The paper also discusses future perspectives on RC systems, including materials, architecture, and applications. In-memory and in-sensor RC systems offer significant energy efficiency and reduced training costs compared to traditional deep neural networks (DNNs). The review highlights the potential of memristive devices in enabling efficient and fast learning for intelligent edge systems.