Motivated by Friedlander-Walker's construction of semi-topological K-theory and Friedlander-Lawson's morphic cohomology, we construct and study an (oriented) cohomology theory defined on smooth complex varieties, called "semi-topological cobordism". The construction combines the semi-topological construction of Friedlander-Walker and motivic stable homotopy theory. One of its features is that with finite coefficients semi-topological cobordism agrees with Voevodsky's algebraic cobordism, on the other hand for certain nice classes of varieties (e.g. smooth cellular varieties) the semi-topological cobordism of X agrees integrally with the topological cobordism of X.