The problem of black body radiation, when measured by a moving observer, has a pivotal role in relativistic thermodynamics. Mutually, it depends on the thermodynamical definition of the thermal equilibrium and temperature of moving bodies, i.e. under a Lorentz transformation, and also in a gravitational field. Surprisingly, even after more than a century, relativistic thermodynamics is not a mature theory and is still an open problem without a consensus. This article is a brief review of the evolution of this theory with a special focus on the black body radiation in moving frames. As an application, we use the results in the most interesting topics of the quantum field theory in curved space: Hawking radiation, and Unruh effect. Note: We recast the adopted discussions of old-style papers in modern terminology , notations, and conventions as far as possible. In particular, we choose the signature (−, +, +, +) for the metric and use the Einstein summation convention. Throughout this paper, γ = (1 − v 2 /c 2) −1/2 is the Lorentz-Fitzgerald contraction factor, and the general-relativistic system of units (c = 1 and G = 1) is opted, unless explicitly stated. The spacetime indices are in Greek letters µ, ν, ... = 0, 1, 2, 3, and the space indices are in Latin letters i, j, k, ... = 1, 2, 3.