The experimental and theoretical study of molecular anions has undergone explosive growth over the past 40 years. Advances in techniques used to generate anions in appreciable numbers as well as new ion-storage, ion-optics, and laser spectroscopic tools have been key on the experimental front. Theoretical developments on the electronic structure and molecular dynamics fronts now allow one to achieve higher accuracy and to study electronically metastable states, thus bringing theory in close collaboration with experiment in this field. In this article, many of the experimental and theoretical challenges specific to studying molecular anions are discussed. Results from many research groups on several classes of molecular anions are overviewed, and both literature citations and active (in online html and pdf versions) links to numerous contributing scientists' Web sites are provided. Specific focus is made on the following families of anions: dipole-bound, zwitterion-bound, double-Rydberg, multiply charged, metastable, cluster-based, and biological anions. In discussing each kind of anion, emphasis is placed on the structural, energetic, spectroscopic, and chemical-reactivity characteristics that make these anions novel, interesting, and important.