Lis­ten­ing to speech is often demand­ing because of sig­nal degra­da­tions and the pres­ence of dis­tract­ing sounds (i.e., “noise”). The ques­tion how the brain achieves the task of extract­ing only rel­e­vant infor­ma­tion from the mix­ture of sounds reach­ing the ear (i.e., “cock­tail par­ty prob­lem”) is still open. In anal­o­gy to recent find­ings in vision, we pro­pose cor­ti­cal alpha (~10 Hz) oscil­la­tions mea­sur­able using M/EEG as a piv­otal mech­a­nism to selec­tive­ly inhib­it the pro­cess­ing of noise to improve audi­to­ry selec­tive atten­tion to task-rel­e­vant sig­nals. We review ini­tial evi­dence of enhanced alpha activ­i­ty in selec­tive lis­ten­ing tasks, sug­gest­ing a sig­nif­i­cant role of alpha-mod­u­lat­ed noise sup­pres­sion in speech. We dis­cuss the impor­tance of dis­so­ci­at­ing between noise inter­fer­ence in the audi­to­ry periph­ery (i.e., ener­getic mask­ing) and noise inter­fer­ence with more cen­tral cog­ni­tive aspects of speech pro­cess­ing (i.e., infor­ma­tion­al mask­ing). Final­ly, we point out the adverse effects of age-relat­ed hear­ing loss and/or cog­ni­tive decline on audi­to­ry selec­tive inhi­bi­tion. With this per­spec­tive arti­cle, we set the stage for future stud­ies on the inhibito­ry role of alpha oscil­la­tions for speech pro­cess­ing in chal­leng­ing lis­ten­ing situations.