Lead halide perovskites have emerged as one of the most promising candidates for photovoltaic and photonic devices in decades. Motivated by the aim to further improve their optical or optoelectronic performance, revealing the carrier kinetics (e.g., carrier diffusion, directional transport and charge separation) within becomes an essential subject to explore. For the carriers in perovskite semiconductors, time-resolved spectroscopy serves as a sharp "eye" for researchers to observe their underlying behaviors. In this Review, we focus on the recent progress in probing and regulating the carrier transport kinetics in perovskite single crystals, polycrystalline thin films, and two-dimensional perovskites, investigated mainly by time-resolved photoluminescence (TR-PL) and ultrafast transient absorption (TA) spectroscopy. Meanwhile, challenges and outlooks in exploring perovskite carrier transport are discussed. With a deeper understanding of carrier dynamics, we are on the way to uncover the photophysical fundamentals for further improvement in optoelectronic performance of perovskite-based devices.