  Following an idea of Kontsevich, we introduce and study the notion of formal
completion of a compactly generated (by a set of objects) enhanced triangulated
category along a full thick essentially small triangulated subcategory.
  In particular, we prove (answering a question of Kontsevich) that using
categorical formal completion, one can obtain ordinary formal completions of
Noetherian schemes along closed subschemes. Moreover, we show that
Beilinson-Parshin adeles can be also obtained using categorical formal
completion.
