Although semiconductor devices have progressively replaced vacuum tubes in nearly all applications, vacuum-tube amplifiers are still in use by professional musicians due to their tonal characteristics. Over the years, many different techniques have been proposed with the goal of reproducing the timbral characteristics of these circuits. This paper presents a review on the methodologies that have been used to emulate tube circuits over the last 30 years for musical applications. The first part of the paper introduces the basic principles of tube circuits, with a common cathode triode example. The remainder of the paper reviews the tube sound simulation devices. The first of these emulations used analog operational amplifier circuits with the negative feedback designed to reproduce tube transfer. As DSP became more popular over the last decades for audio applications, efforts towards digital tube circuit simulation algorithms were initiated. Simulation of these devices are basically divided into linear models with digital filters that correspond to IIR analog filters and nonlinear digital models that corresponds to the tube circuit itself. The simulation of the first is straightforward, normally accomplished by the use of FIR digital filters. The last can be either accomplished approximation equations, that are known as digital waveshapers and their variants or by circuit derived techniques, such as the resolution of circuit ordinary differential equations solvers. Wave digital filter models are also variants of circuit simulation techniques that are also treated in this paper. The circuit derived techniques yield more precise simulations over the waveshapers but are always computationally more expensive so that a compromise between accuracy and efficiency is needed for real-time simulation of these devices.