An accurate estimate of the theoretical power limit of turbines in free fluid flows is important because of growing interest in the development of wind power and zero-head water power resources. The latter includes the huge kinetic energy of ocean currents, tidal streams, and rivers without dams. Knowledge of turbine efficiency limits helps to optimize design of hydro and wind power farms. An explicitly solvable new mathematical model for estimating the maximum efficiency of turbines in a free (nonducted) fluid is presented. This result can be used for hydropower turbines where construction of dams is impossible (in oceans) or undesirable (in rivers), as well as for wind power farms. The model deals with a finite two-dimensional, partially penetrable plate in an incompressible fluid. It is nearly ideal for two-dimensional propellers and less suitable for three-dimensional cross-flow Darrieus and helical turbines. The most interesting finding of our analysis is that the maximum efficiency of the plane propeller is about 30 percent for free fluids. This is in a sharp contrast to the 60 percent given by the Betz limit, commonly used now for decades. It is shown that the Betz overestimate results from neglecting the curvature of the fluid streams. We also show that the three-dimensional helical turbine is more efficient than the two-dimensional propeller at least in water applications. Moreover well-documented tests have shown that the helical turbine has an efficiency of 35 percent, making it preferable for use in free water currents.