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The Nyström method (NM) is used to solve for electromagnetic scattering by 3-D composite objects based on surface integral equations (SIEs). These SIEs include both equivalent electric and magnetic currents as unknowns since composite media exist. In the method-of-moments (MoM) solution for these SIEs, one may encounter the problem of how to represent the magnetic current using an appropriate basis function if the electric current is represented by Rao-Wilton-Glisson (RWG) basis function. Some choices like RWG, n̂ × RWG, or dual basis function in representing the magnetic current may have the instability, fictitious charge, or high-cost problems, respectively, and thus, are not ideal. Compared with the MoM, the NM is simpler to implement, and most importantly, it can get rid of these problems. We employ this method to solve the E-field, H-field, and combined-field SIEs with efficient local correction schemes. Numerical examples show that the NM can give stable and efficient solutions for both near and far fields, when away from the resonant frequencies in E-field and H-field formulations, even for relatively complicated structures.