$A_\infty$-categories basic reference Can anyone provide me with a basic reference on $A_\infty$ categories?  
 A: I am currently trying to learn this, and this paper of B. Keller proved very useful. This one of J. Huebschmann seems a bit less basic, but might be usefull too. I must say I'm very interested in any other answers.
A: I believe the thesis of Kenji Lefèvre-Hasegawa is a remarkable piece of work, and is very readable (references spotted by Samuel Tinguely are very good, but they are about $A_\infty$-algebras): 
http://www.math.jussieu.fr/~keller/lefevre/TheseFinale/tel-00007761.pdf
Unfortunately it is in only available in French. 
A: To me the best path to $A_\infty$-categories is via topology. One first gets familiar with the notion of $A_\infty$-space as a natural generalization of that of topological monoid (the classical reference by Jim Stasheff is probably still where one should have a look for a complete account on this). Next one considers topological categories as a natural generalization of topological monoids (the only difference being that the product is not defined on $M\times M$ but on a fibered product $M_1\times_{M_0}M_1$, the latter reducing to the first for the space of objects $M_0$ consisting of a single point). Then one mixes these two generalizations of topological monoid and gets the definition of "$A_\infty$ topological category" in the most natural possible way (in my opinion). Once one is familiar with this, one sees that nothing changes if instead of being in the topological setting one works in any setting where "homotopies" are meaningful. For instance one can work with dg-categories, and this gives the version of $A_\infty$-category one usually meets.
A: you will like this one, i believe. Have fun!
http://arxiv.org/abs/math/0306332
by the way, a short notes:
http://arxiv.org/abs/1310.3718
A: Here's a list of references:

*

*BLM,              483 Pages, Pretriangulated _∞-Categories;

*COS,               25 Pages, Localizations of the Category of _∞-Categories and Internal Homs;

*Faonte,           156 Pages, Nerve construction, _∞-functors and homotopy theory of differential graded categories;

*Faonte,            47 Pages, Simplicial nerve of an _∞-category;

*Faonte,            53 Pages, _∞-functors and homotopy theory of dg-categories;

*Fukaya,            97 Pages, Floer Homology and Mirror Symmetry;

*Horel,             10 Pages, The homotopy theory of _∞-categories;

*Keller,            31 Pages, Introduction to _∞-algebras and modules;

*KS,                70 Pages, Notes on _∞-algebras, _∞-categories and non-commutative geometry I;

*Lefèvre-Hasegawa, 230 Pages, Sur les _∞-Catégories;

*Ornaghi,          113 Pages, Comparison Results About DG-Categories, _∞-Categories, Stable ∞-Categories and Noncommutative Motives;

*Seidel,           334 Pages, Fukaya Categories and Picard–Lefschetz Theory.

