Microwave signal filtering is a fundamental and central functionality in radio-frequency (RF) systems. Underpinned by advanced integrated photonics technologies, emerging integrated microwave photonic (IMWP) filter platforms enable reconfigurable and widely tunable RF signal filtering functionalities that were unattainable using conventional electronics while also exhibiting superior features in terms of compactness, light weight, stability, low power consumption, and low latency. This paper presents a comprehensive review of the principles, architectures, and performance of IMWP filters. We highlight recent advances of IMWP filters enabled by on-chip nonlinear optics, RF-interference technology and emerging integration platforms, with an emphasis on the RF performance which is critical for their usability in real-world applications. We conclude with a perspective on future research challenges and new possibilities for IMWP filters.