This paper introduces a power-law subordinator and a power-law Lévy process whose Laplace transform and characteristic function are simply made up of power functions or the ratio of power functions, respectively, and proposes a power-law vector random field whose finite-dimensional characteristic functions consist merely of a power function or the ratio of two power functions. They may or may not have first-order moment, and contain Linnik, variance Gamma, and Laplace Lévy processes (vector random fields) as special cases. For a second-order power-law vector random field, it is fully characterized by its mean vector function and its covariance matrix function, just like a Gaussian vector random field. An important feature of the power-law Lévy processes (random fields) is that they can be used as the building blocks to construct other Lévy processes (random fields), such as hyperbolic secant, cosine ratio, and sine ratio Lévy processes (random fields).