Vertex (operator) algebras have seen numerous applications in algebraic geometry, particularly in the study of moduli spaces of curves. Over the years a number of equivalent algebraic structures to vertex algebras have seen higher-dimensional generalizations, but there have been very few attempts for vertex algebras. It is a natural question as to whether there exists a generalization of vertex algebras that plays a similar role in the study of moduli spaces of higher-dimensional varieties. We propose one candidate, which we call Cohomological Vertex Algebras. These structures are formed from vertex algebras by replacing the role of global sections of the formal punctured 1-disk (over which the local observables of the vertex algebra live) with the full cohomology of the formal punctured n-disk. We develop the theory from basic definitions entirely parallel to that of the one-dimensional theory, providing analogous results such as a version of the Jacobi identity, the reconstruction theorem, which is our primary method of producing examples, and the interpretation of vertex algebras in terms of correlation functions. Further we describe related structures such as vertex Lie algebras, vertex Poisson algebras, universal enveloping algebras, and higher Zhu algebras.