The flow through the air turbine of an oscillating water column (OWC) wave energy converter is reciprocating and is random and highly variable. It is not surprising that the time-averaged efficiency of the air turbine is substantially lower than that of a conventional turbine working in nearly steady conditions. A new type of radial-flow self-rectifying turbine (named here biradial turbine) is described in the paper. The two inlet/outlet openings of the rotor are axially offset from each other and face radially the surrounding space. The turbine is symmetrical with respect to a plane perpendicular to its axis of rotation. The rotor blades are surrounded by a pair of radial-flow guide-vane rows. Each guide vane row is connected to the rotor by an axisymmetric duct whose walls are flat discs. A two-dimensional flow method is used first as a preliminary design tool for the turbine geometry. More detailed numerical results are then obtained with the aid of a commercial three-dimensional real-fluid CFD code, which allows a more refined geometry optimization to be carried out, and yields results for flow details through the turbine and for the turbine overall performance under several operating conditions.