<p>We investigate how to give new representations of (<i>b</i>, <i>c</i>)-inverses in terms of core inverses and Drazin inverses. Various new criteria and representations of (<i>b</i>, <i>c</i>)-inverses in terms of core inverses and Drazin inverses are established from a new perspective. Since Moore-Penrose inverses, pseudo-inverses, core inverses, dual core inverses, and Bott-Duffin (<i>e</i>, <i>f</i>)-inverses are the special cases of (<i>b</i>, <i>c</i>)-inverses, new characterizations involving these generalized inverses are also established as corollaries of our results.</p>