Scene representation is a crucial design choice in robotic manipulation systems. An ideal representation is expected to be 3D, dynamic, and semantic to meet the demands of diverse manipulation tasks. However, previous works often lack all three properties simultaneously. In this work, we introduce D$^3$Fields -- dynamic 3D descriptor fields. These fields are implicit 3D representations that take in 3D points and output semantic features and instance masks. They can also capture the dynamics of the underlying 3D environments. Specifically, we project arbitrary 3D points in the workspace onto multi-view 2D visual observations and interpolate features derived from visual foundational models. The resulting fused descriptor fields allow for flexible goal specifications using 2D images with varied contexts, styles, and instances. To evaluate the effectiveness of these descriptor fields, we apply our representation to rearrangement tasks in a zero-shot manner. Through extensive evaluation in real worlds and simulations, we demonstrate that D$^3$Fields are effective for zero-shot generalizable rearrangement tasks. We also compare D$^3$Fields with state-of-the-art implicit 3D representations and show significant improvements in effectiveness and efficiency.