The theory of sinusoidal alternating field losses in a cylindrical multifilamentary superconducting wire is discussed. Expressions for the losses in a wire of radius Ro, twist length , and skin depth are presented for an alternating magnetic field applied transverse to the wire axis. Eddy current and hysteresis loss expressions are presented for the cases of (1) low frequency losses ( ), (2) shielding from surface currents ( ) and (3) skin effect losses ( ). Hysteresis loss expressions are presented for the individual filament losses in the interior of the wire when these filaments are completely penetrated, and for the loss occurring in an outer current-saturated layer of the wire. The loss expressions presented were determined assuming a continuum model with anisotropic conductivity for the superconductor. The case of a permeability different from unity, applicable for weak fields, is discussed.