The process of converting analog signals, or continuous functions, to digital signals is a classical problem in signal processing. Many analog to digital converters follow the paradigm of taking lots of samples and then compressing afterwards. One might wonder if you could be more prudent and only take as many samples as you'd need to guarantee that you can reconstruct a given signal. That is, can you compress while simultaneously measuring a signal? Compressed sensing's hallmark achievement is proving that for a large class of structured signals this is indeed possible. In this talk, I will introduce compressed sensing by defining a mathematical model for signal acquisition and outline procedures that guarantee signal reconstruction. There will be lots of pictures, and lots of solving $Ax=b$