This paper studies Dehn surgery on a large class of links, called arborescent links. It will be shown that if an arborescent link L is sufficiently complicated, in the sense that it is composed of at least 4 rational tangles T (p i /q i ) with all q i > 2, and none of its length 2 tangles are of the form T (1/2q 1 , 1/2q 2 ), then all complete surgeries on L produce Haken manifolds. The proof needs some result on surgery on knots in tangle spaces. Let T (r/2s, p/2q) = (B, t 1 ∪ t 2 ∪ K) be a

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... gle with K a closed circle, and let M = B − Int N (t 1 ∪ t 2 ). We will show that if s > 1 and p ≡ ±1 mod 2q, then ∂M remains incompressible after all nontrivial surgeries on K. Two bridge links are a subclass of arborescent links. For such a link L(p/q), most Dehn surgeries on it are non-Haken. However, it will be shown that all complete surgeries yield manifolds containing essential laminations, unless p/q has a partial fraction decomposition of the form 1/(r − 1/s), in which case it does admit non-laminar surgeries.