In statistical mechanics, entropy is related to the number of states with the same energy, which indicates that entropy reflects a property related to the information and randomness of the system. The concept of entropy is extended to non-equilibrium states, where it is defined as a function of the distribution function of states (Boltzmann-Gibbs entropy, Shannon entropy). In quantum mechanics, entropy as a function of state distribution is defined using the density matrix, known as von Neumann entropy. In stationary states, it reproduces the thermodynamic entropy. Moreover, the relative entropy, known as the Kullback-Leibler divergence (KLD), provides a description of how the system evolves toward its equilibrium state. The idea of so-called entanglement entropy of quantum ground states in interacting systems has been introduced, which characterizes quantum mechanical correlations within the system. The entropy of black holes is briefly reviewed, and entropy is also extended to non-extensive systems, where generalized forms of entropy, such as Tsallis entropy, are introduced.