This paper presents a method for efficiently predicting many properties of a quantum system using very few measurements. The method involves creating a classical shadow, a classical description of a quantum state derived from random measurements. This shadow can be used to predict various properties of the quantum state, including quantum fidelities, entanglement entropies, two-point correlation functions, and expectation values of local observables. The number of measurements required is independent of the system size and saturates information-theoretic lower bounds. The method is supported by extensive numerical experiments and is shown to be more efficient than existing methods, including machine learning approaches. The classical shadow is constructed by applying random unitary transformations and measuring the resulting states in the computational basis. The classical shadow can be used to predict linear functions of the quantum state using a median-of-means protocol. Theoretical guarantees are provided for the performance of the method, showing that a logarithmic number of measurements suffices to predict a large number of target functions. The method is also shown to be optimal in terms of information-theoretic lower bounds. The paper also discusses the application of classical shadows to predicting nonlinear functions and the limitations of the method for certain types of observables. The results demonstrate that classical shadows can be used to efficiently predict many properties of quantum systems, making them a promising tool for quantum technologies.This paper presents a method for efficiently predicting many properties of a quantum system using very few measurements. The method involves creating a classical shadow, a classical description of a quantum state derived from random measurements. This shadow can be used to predict various properties of the quantum state, including quantum fidelities, entanglement entropies, two-point correlation functions, and expectation values of local observables. The number of measurements required is independent of the system size and saturates information-theoretic lower bounds. The method is supported by extensive numerical experiments and is shown to be more efficient than existing methods, including machine learning approaches. The classical shadow is constructed by applying random unitary transformations and measuring the resulting states in the computational basis. The classical shadow can be used to predict linear functions of the quantum state using a median-of-means protocol. Theoretical guarantees are provided for the performance of the method, showing that a logarithmic number of measurements suffices to predict a large number of target functions. The method is also shown to be optimal in terms of information-theoretic lower bounds. The paper also discusses the application of classical shadows to predicting nonlinear functions and the limitations of the method for certain types of observables. The results demonstrate that classical shadows can be used to efficiently predict many properties of quantum systems, making them a promising tool for quantum technologies.