Cheuk laser-cools and traps individual laser-coolable molecules (e.g. CaF) in optical tweezer arrays, achieving high-fidelity non-destructive imaging, Raman sideband cooling, and on-demand entanglement of molecular qubits, with explicit applications to quantum simulation, quantum information processing, and quantum-enhanced sensing/precision measurement. The rich internal structure of molecules gives access to new sensing modalities (e.g. searches for new physics) that complement atom-based quantum sensors.