This Perspective provides an overview of state-of-the-art ab initio quantum chemical methodology and applications. The methods that are discussed include coupled cluster theory, localized second-order Moller-Plesset perturbation theory, multireference perturbation approaches, and density functional theory. The accuracy of each approach for key chemical properties is summarized, and the computational performance is analyzed, emphasizing significant advances in algorithms and implementation over the past decade. Incorporation of a condensed-phase environment by means of mixed quantum mechanical͞molecular mechanics or self-consistent reaction field techniques, is presented. A wide range of illustrative applications, focusing on materials science and biology, are discussed briefly.coupled cluster ͉ density functional theory ͉ second-order Modler-Plesser perturbation theory ͉ mixed quantum͞molecular mechanics O ver the past three decades, ab initio quantum chemistry has become an essential tool in the study of atoms and molecules and, increasingly, in modeling complex systems such as those arising in biology and materials science. The underlying core technology is computational solution of the electronic Schrodinger equation; given the positions of a collection of atomic nuclei, and the total number of electrons in the system, calculate the electronic energy, electron density, and other properties by means of a well defined, automated approximation (a ''model chemistry''). The ability to obtain ''goodenough'' solutions to the electronic Schrodinger equation for systems containing tens, or even hundreds, of atoms has revolutionized the ability of theoretical chemistry to address important problems in a wide range of disciplines; the Nobel Prize awarded to John Pople and Walter Kohn in 1998 is a reflection of this observation.In its exact form, the electronic Schrodinger equation is a many-body problem, whose computational complexity grows exponentially with the number of electrons, and hence, a brute force solution is intractable. Hartree-Fock theory, a mean field approach, produces reasonable results for many properties but is incapable of providing a robust description of reactive chemical events in which electron correlation has a major role. Thus, a key problem has been the development of treatments of electron correlation that exhibit a tractable scaling in computational effort with the size of the system. The considerable progress that has been made along these lines is outlined below.Given a well defined theoretical framework of approximation, the next requirement is efficient computational implementation. Considerable sophistication is required to achieve acceptable accuracy and efficiency; the leading quantum chemistry programs are millions of lines of computer code, and mathematical algorithms to reduce formal scaling of computational effort with system size have an increasingly crucial role in meeting the challenge of handling complex system relevant to practical applications. Below, the most important comp...