In this chapter we initially present techniques for determining the behavior of solutions to systems of first-order differential equations without first finding the solutions. To this end, we define and discuss equilibrium points (critical points), various types of stability and instability, and phase-plane graphs. Next, we show how to use computer software to solve systems of first-order differential equations numerically, how to graph the solution components, and how to produce phase-plane graphs. We also state stability theorems for systems of first-order differential equations. Throughout this chapter we develop and discuss a wide variety of models and applications which can be written as vector initial value problems and then solved numerically.