Abstract: To any compact Riemann surface one can associate (at least) two kinds of cohomological Hall algebras: one whose underlying vector space is the (singular) cohomology of the moduli stack of vector bundles (or coherent sheaves) on X, and one whose underlying vector space is the (singular) cohomology of the moduli stack of Higgs bundles (or Higgs sheaves) on X. We will describe these algebras and give some applications and conjectures.