In this talk we introduce and study a new property of infinite words: An infinite word x on an alphabet A is said to be it self-shuffling, if x admits factorizations: $x=prod_{i=1}^infty U_iV_i=prod_{i=1}^infty U_i=prod_{i=1}^infty V_i$ with $U_i,V_i in A^*$. In other words, there exists a shuffle of x with itself which reproduces x. This property of infinite words is shown to be an intrinsic property of the word and not of its language (set of factors). For instance, every aperiodic uniformly recurrent word contains a non self-shuffling word in its shift orbit closure. On the other hand, we build an infinite word such that no word in its shift orbit closure is self-shuffling. We prove that many important and well studied words are self-shuffling: This includes the Thue-Morse word and all Sturmian words (except those of the form aC where a ∈ {0,1} and C is a characteristic Sturmian word). We further establish a number of necessary conditions for a word to be self-shuffling, and show that certain other important words (including the paper-folding word and infinite Lyndon words) are not self-shuffling. One important feature of self-shuffling words is its morphic invariance: The morphic image of a self-shuffling word is again self-shuffling. This provides a useful tool for showing that one word is not the morphic image of another. In addition to its morphic invariance, this new notion has other unexpected applications: For instance, as a consequence of our characterization of self-shuffling Sturmian words, we recover a number theoretic result, originally due to Yasutomi, on a classification of pure morphic Sturmian words in the orbit of the characteristic. Finally, we provide a positive answer to a recent question by T. Harju whether square-free self-shuffling words exist and discuss self-shufflings in a shift orbit closure. E. Charlier, T. Kamae, S. Puzynina, L. Q. Zamboni: Infinite self-shuffling words. J. Comb. Theory, Ser. A 128: 1-40 (2014) M. Müller, S. Puzynina, M. Rao: On Shuffling of Infinite Square-Free Words. Electr. J. Comb. 22(1): P1.55 (2015)