Some permanents of circulant (0,1) matrices are computed. Three methods are used. First, the permanent of a Kronecker product is computed by directly counting diagonals. Secondly, Lagrange expansion is used to calculate a recurrence for a family of sparse circulants. Finally, a "complement expansion" method is used to calculate a recurrence for a permanent of a circulant with few zero entries. Also, a bound on the number of different permanents of circulant matrices with a given row sum is obtained.