Logarithmic Gromov-Witten theory virtually counts the number of holomorphic curves with prescribed tangency condition along boundary divisors. In this talk I will introduce a variant of logarithmic maps called the punctured logarithmic maps. They naturally appear in a generalization of the gluing formulas of Li-Ruan and Jun Li. The punctured invariants play the role of relative invariants in these classical gluing formulas. They extend logarithmic Gromov-Witten theory by allowing negative tangency conditions with boundary divisors. This talk is based on a joint work with Dan Abramovich, Mark Gross and Bernd Siebert.