The Ward equation, also called the modified 2+1 chiral model,is obtained by a dimension reduction and a gauge fixing from the
self-dual Yang-Mills field equation on R2,2. It has a Lax pair and is an integrable system. Ward constructed solitons whose ex-
tended solutions have distinct simple poles. He also used a limiting method to construct 2-solitons whose extended solutions have a
double pole. Ioannidou and Zakrzewski, and Anand constructed more soliton solutions whose extended solutions have a double or
triple pole. Some of the main results of this paper are: (i) We construct algebraic B¨acklund transformations (BTs) that generate new solutions of the Ward equation from a given one by an algebraic method. (ii) We use an order k limiting method and algebraic BTs to construct explicit Ward solitons, whose extended solutions have arbitrary poles and multiplicities. (iii) We prove
that our construction gives all solitons of the Ward equation explicitly and the entries of Ward solitons must be rational functions
in x, y and t. (iv) Since stationary Ward solitons are unitons, our method also gives an explicit construction of all k-unitons from finite sequences of rational maps from C to Cn.