We discuss extension of soliton theory and integrable systems to non-commutative (NC) spaces, focusing on integrable aspects of non-commutative anti-self-dual Yang-Mills (ASDYM) equations. We present Backlund transformations for the G=U(2) non-commutative anti-self-dual Yang-Mills equations and give wide class of exact solutions of them (not only instanton-type solutions with finite action). We find that one kind of non-commutative determinants,quasi-determinants, play crucial roles in the construction of non-commutative solutions. We also discuss twistor descriptions of the results, analysis of the exact solutions, and reduction of a non-commutative anti-self-dual Yang-Mills equation to non-commutative integrable equations such non-commutative KdV and Toda equations, if possible. This is partially based on collaboration with C. Gilson and J. Nimmo (Glasgow).