We study CappelFs UNil group, UNÜ2 n (R\^\^-^ for any ring and pair of bimodules with Involution (R; ^I 9^_l ). We show that, in the geometrically significant cases, this group is isomorphic to the Wall-Ranicki L-group, £ e (A a [f]), for a certain additive polynomial extension category A a [f ]. We then introduce an Arf invariant for UNH 2n (R', R, R) when the involution is trivial. We use this to compute UNil^R', R, R) when R is a Dedekind domain in which 2 is prime. We also show that for a suitable choice of (A, a), the Nil group of (A, a) coincides with the Nil group of Bass-Farrell and with the Nil group of Waldhausen. 1991 Mathematics Subject Classification: 57N15, 57R67. § 1. Introduction 1.0 The primary goal of this paper is to analyze Cappell's group UNil* (R\ 3# l9 $ -±), n > 0 äs defined in [Cl], (See also [Rl] Chapter 7). In particular, we calculate UNÜ2" (R', R, R) for any semisimple ring, and any Dedekind domain in which 2 is prime. Our analysis inevitably also draws in the Nil groups defined by Waldhausen [Wl], [W2], [W3], (also see [C4] p. 125), by Farrell [Fl], and by Bass [B]. It leads to interesting new interpretations of these äs well. The main results are 2.11, 3.9,4.9, 6.1, and 6.2. We summarize our results in l .2 below. But first we want to explain why a better underStanding of UNil seems so important to us.1.1 Motivation. Suppose/: M -+ X is a homotopy equivalence of compact closed manifolds, and X is the union of two other manifolds, X=X i uX_ i9 with X 0 = dX t = dX_ x = n X_ 1 . Assume that X and X 0 are connected, and that the fundamental groups of X i9 X-l9 X, and X 0 are denoted G