  Let (M,w) be a compact symplectic manifold, and L a compact, embedded
Lagrangian submanifold in M. Fukaya, Oh, Ohta and Ono construct Lagrangian
Floer cohomology for such M,L, yielding groups HF^*(L,b;\Lambda) for one
Lagrangian or HF^*((L,b),(L',b');\Lambda) for two, where b,b' are choices of
bounding cochains, and exist if and only if L,L' have unobstructed Floer
cohomology. These are independent of choices up to canonical isomorphism, and
have important invariance properties under Hamiltonian equivalence. Floer
cohomology groups are the morphism groups in the derived Fukaya category of
(M,w), and so are an essential part of the Homological Mirror Symmetry
Conjecture of Kontsevich.
  The goal of this paper is to extend all this to immersed Lagrangians L in M
with immersion i : L --> M, with transverse self-intersections. In the embedded
case, Floer cohomology HF^*(L,b;\Lambda) is a modified, 'quantized' version of
cohomology H^*(L;\Lambda) over the Novikov ring \Lambda. In our immersed case,
HF^*(L,b;\Lambda) turns out to be a quantized version of the sum of
H^*(L;\Lambda) with a \Lambda-module spanned by pairs (p,q) for p,q distinct
points of L with i(p)=i(q) in M.
  The theory becomes simpler and more powerful for graded Lagrangians in
Calabi-Yau manifolds, when we can work over a smaller Novikov ring
\Lambda_{CY}. The proofs involve associating a gapped filtered A-infinity
algebra over \Lambda or \Lambda_{CY} to i : L --> M, which is independent of
nearly all choices up to canonical homotopy equivalence, and is built using a
series of finite approximations called A_{N,0} algebras for N=0,1,2,...
