TITLE: $A_\infty$-categories basic reference
QUESTION [14 upvotes]: Can anyone provide me with a basic reference on $A_\infty$ categories?

REPLY [3 votes]: 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.