In recent studies of art historical methodology, such as Critical Terms for Art History and The Art of Art History, subjectivity, identity, abjection, and other terms have been placed safely in the genealogy of contemporary art history. This paper questions the contemporaneity in the story of contemporary art in our time in relation to two other critical terms that have been regularly cited by contemporary critics, not only in Euro-American fields but also in Korea. The terms are postmedium and postproduction, respectively, as used by Rosalind Krauss and Nicolas Bourriaud. This paper stems from the critical condition in which art criticism and theory have their power in the rise of neo-liberalism. But this paper does not deal with the contemporary as a chronological term for art history but rather examines the three critical terms-contemporaneity, post-medium, and postproduction-that have garnered scholarly attention. I would like to put aside postmodernism for the moment; I don't disregard the postmodern condition although the death of postmodern critical terms has resulted in the loss of its polemical power in art worlds such as in exhibitions, etc. To look at "the postproduction in the age of post-medium age after postmodernism," I first explore Krauss's notion of post-medium because, unlike media artists like Lev Manovich and Peter Weibel, Krauss's post-medium condition is different and insists on medium specificity. In this sense, Krauss has turned out to be another Greenberg in disguise. For her, photography and video are expanded mediums after Greenberg, because Krauss has spent her life explicating those mediums. Under the Cup, her recent publication, came out in 2011, and discusses her desire to defend medium-specificity against the intermedia of installation art found ubiquitously in international exhibitions and biennales. Her usage of post-medium has been taken up by Weibel as postmedia in a broader sense. But whether the post-medium condition or the postmedia age, we nonetheless enter the new age of the contemporary. Consequently, this paper questions what constitutes contemporaneity in our times. It is said that there is nothing new on earth, yet I find original artistic strategies among the younger generation in the postmedia age. The contemporary justifies its place in art fields and criticism by keeping its distance from postmodernism although we still find the remnants of postmodern artistic practices and theoretical foundations. By looking at materials written by Terry Smith, I would like to examine contemporaneity as a rhetoric where artists, critics, and curators endeavor to set up a new spirit of criticism, distant from the past of modernism and postmodernism. In discussions, modernism and postmodernism act as catalysts interacting with each other while justifying their own place. In conclusion, my paper reaches to delineate where the contemporary finds its place among artists' responses and working methods. It explores the postproduction of the Internet and the World Wide Web generations, where images become data rather than representation (of modernism) and appropriation (of postmodernism). This paper analyzes Bourriaud's text, as well as relevant artists like Pierre Huyghe, Liam Gillick, and others. By examining the aforementioned critical terms, I would like to reconsider our own contemporary art in Korea, especially among young artists influenced by digital media and the World Wide Web in the 1990s.