A Mach probe (MP) is an electric probe system to deduce the plasma flow velocity from the ratio of ion saturation currents. Generally, a typical MP is composed of two directional electric probes located at opposite sides of an insulator, which is mostly used as a parallel MP, but there are other MPs such as perpendicular MP (PMP), Gundestrup probe (GP) or rotating probe (RP), and visco-MP (VMP), depending on the shape of the probe holder, location of different probes or the method of collecting ions. For the parallel MP (to be called simply an MP), the relation between the ratio of the upstream ion saturation current density (Jup) to the downstream (Jdn) and the normalized drift velocity (M∞ = vd/√Te/mi) of the plasma has generally been fitted into an exponential form (R = Jup/Jdn ≈ exp[KM∞]). For the GP or RP, with oblique ion collection, the relation becomes R = exp[K(M∞ −M⊥ cot θ)], where K = 2.3~2.5, M∞ is the normalized parallel flow, and M⊥ is the normalized perpendicular flow to the magnetic field, and θ is the angle between the magnetic field and the probe surface. The normalized drift velocity of flowing plasmas is deduced from the ratio (Rm) measured by an MP as M∞ = ln[Rm]/K, where K is a calibration factor depending on the magnetic flux density, collisionality of charged particles and neutrals, viscosity of plasmas, ion temperature, etc. Existing theories of MPs in unmagnetized and magnetized flowing plasmas are introduced in terms of kinetic, fluid and particle-in-cell models or self-consistent and self-similar methods along with key physics and comments. Experimental evidence of relevant models is shown along with validity of related theories. For probes other than the typical parallel MP, the relation between the ratio of ion saturation currents and M∞ can be expressed as a combination of the functional forms: exponential and/or polynomial form of M∞ for PMP; two Rs of two separate MPs for VMP. Collisions of ions/electrons/neutrals, asymmetries of ion temperatures and the existence of hyperthermal electrons, existence of ion beam, supersonic flow and negative ions can affect the deduction of flow velocities by an MP.