  Molecular wires (MW) are the fundamental building blocks for molecular
electronic devices. They consist of a molecular unit connected to two continuum
reservoirs of electrons (usually metallic leads). We rely on Landauer theory as
the basis for studying the conductance properties of MW systems. This relates
the lead to lead current to the transmission probability for an electron to
scatter through the molecule. Two different methods have been developed for the
study of this scattering. One is based on a solution of the Lippmann-Schwinger
equation and the other solves for the {\bf t} matrix using Schroedinger's
equation. We use our methodology to study two problems of current interest. The
first MW system consists of 1,4 benzene-dithiolate (BDT) bonded to two gold
nanocontacts. Our calculations show that the conductance is sensitive to the
chemical bonding between the molecule and the leads. The second system we study
highlights the interesting phenomenon of antiresonances in MW. We derive an
analytic formula predicting at what energies antiresonances should occur in the
transmission spectra of MW. A numerical calculation for a MW consisting of
filter molecules attached to an active molecule shows the existence of an
antiresonance at the energy predicted by our formula.
