This paper discusses various theorems on the approximation capabilities of neural networks (NNs), which are known as universal approximation theorems (UATs). The paper gives a systematic overview of UATs starting from the preliminary results on function approximation, such as Taylor's theorem, Fourier's theorem, Weierstrass approximation theorem, Kolmogorov - Arnold representation theorem, etc. Theoretical and numerical aspects of UATs are covered from both arbitrary width and depth.