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Quantum sensing and metrology use coherent superposition states of quantum systems to detect and measure physical effects of interest. Their sensitivity is typically limited by the standard quantum limit, which bounds the achievable precision in measurements involving nominally identical but uncorrelated quantum systems. Fully quantum metrology involves entanglement in an array of quantum systems, enabling uncertainty reduction below the standard quantum limit. Although ultracold atoms have been widely used for applications such as atomic clocks or gravitational sensors, molecules show higher sensitivity to many interesting phenomena, including the existence of new, symmetry-violating forces mediated by massive particles. Recent advancements in molecular cooling, trapping and control techniques have enabled the use of molecules for quantum sensing and metrology. This Review describes these advancements and explores the potential of the rich internal structure and enhanced coupling strengths of molecules to probe fundamental physics and drive progress in the field.