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Can one use a custom stylesheet in Wolfram Cloud? I didn't see such an option but maybe there's a non-obvious way.

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They work, but you need to a) write stylesheet notebook expression or b) edit in desktop > upload and then set StyleDefinitions programmatically.

Additionally SetOptions[EvaluationNotebook[], StyleDefinitions -> ...] may not refresh styles automatically but if you reopen the notebook it should be ok.

Try it on a saved cloud notebook and reopen it:

SetOptions[EvaluationNotebook[],
  StyleDefinitions->Notebook[{
    Cell[StyleData[StyleDefinitions->"Default.nb"]],
    Cell[StyleData["Notebook"],Background->Red],
    Cell[StyleData["Input"],CellFrame->True]
  }]
]

enter image description here

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