Broadband electromagnetic induction (EMI) methods are promising in the detection and discrimination of subsurface metallic targets. We compute EMI responses from conducting and permeable spheroids by using a field expansion method which is based on the separation of variables in spheroidal coordinates. In addition to an exact formulation which utilizes the vector spheroidal wavefunctions inside the spheroid, we also develop an approximate theory known as the small penetration-depth approximation (SPA). For general permeability, SPA is applicable at high frequency and compliments the exact formulation which breaks down at high frequency. However, when the permeability of the spheroid is large enough, the SPA yields an accurate broadband response. Numerical results for the far-field frequency responses from prolate and oblate spheroids are presented. By neglecting mutual interactions between the spheroids, we also study the broadband EMI response from a collection of spheroids that are randomly oriented and have different sizes.