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I will discuss my most recent result that for an arbitrary vertex operator algebra, or more generally, a grading-restricted Möbius vertex algebra V, (logarithmic) intertwining operators among C_1-cofinite grading-restricted generalized V-modules satisfy the associativity property (operator product expansion) and the category of C_1-cofinite grading-restricted generalized V-modules has natural vertex and braided tensor category structures. The proof and construction in this work is based on a generalization of the Huang-Lepowsky-Zhang logarithmic tensor category theory to the case that the category might not be closed under the contragredient functor. The result follows after the assumptions to use this generalization are all verified.