This paper presents a new type of distributed dexterous manipulator: delta arrays. Our delta array setup consists of 64 linearly-actuated delta robots with 3D-printed compliant linkages. Through the design of the individual delta robots, the modular array structure, and distributed communication and control, we study a wide range of in-plane and out-of-plane manipulations, as well as prehensile manipulations among subsets of neighboring delta robots. We also demonstrate dexterous manipulation capabilities of the delta array using reinforcement learning while leveraging compliance. Our evaluations show that the resulting 192 DoF compliant robot is capable of performing various coordinated distributed manipulations of a variety of objects, including translation, alignment, prehensile squeezing, lifting, and grasping.