In his personal retrospective on his work in mathematics, Grothendieck said that of all the objects he discovered, that motives were "the most fascinating, the most charged with mystery, and were at the very heart of the deep identity between 'geometry' and 'arithmetic.'" Even today, there is no widely accepted definition of what a motive is (although there are many things that we recognize as “motivic”), but the “yoga of motives” has been a guiding force in mathematical investigations for decades. We’ll try and explain in concrete terms where such objects arise, and give examples of how this perspective can be applied to understand a number of classical mathematical questions (which make no mention of the word “motive”).