R. de la Madrid The Importance of Boundary Conditions in Quantum Mechanics (102K, Gzipeed tar file) ABSTRACT. We discuss the role of boundary conditions in determining the physical content of the solutions of the Schr\"odinger equation. We study the standing-wave, the ``in,'' the ``out,'' and the purely outgoing boundary conditions. As well, we rephrase Feynman's $+i \varepsilon$ prescription as a time-asymmetric, causal boundary condition, and discuss the connection of Feynman's $+i \varepsilon$ prescription with the arrow of time of Quantum Electrodynamics. A parallel of this arrow of time with that of Classical Electrodynamics is made. We conclude that in general, the time evolution of a closed quantum system has indeed an arrow of time built into the propagators.