Integrated photonics, where optical components are fabricated on a chip-scale platform leveraging standard microfabrication technologies, has transformed telecommunications and data communications, quantum optics, and molecular sensing. Optical spectrometry is yet another field that integrated photonics is poised to revolutionize. Unlike traditional bulky, costly benchtop spectrometers, integrated photonics promises miniaturized, rugged, and low-cost spectrometer-on-a-chip modules with broad application prospects ranging from communications to medical imaging. In this review, we survey the various designs of integrated photonic spectrometers through the lens of their underlying operating principles, aiming to reveal quantitative performance scaling laws that transcend specific implementations. This approach enables a general, physically grounded comparison of spectrometer capabilities without being bogged down by device-level details. We further provide guidance on selecting appropriate spectrometer architectures for different applications, taking into account not only their reported advantages but also the practical limitations and implementation challenges.