The notion of a compact operator is an extension of the class of finite rank operators. They behave more like matrices than general operators. They have a structure theory which is suggested by results in linear algebra, although there are differences. We also introduce the notion of Fredholm operators, which are invertible modulo compact operators. The spectral theorem for compact normal operators on Hilbert space is also established. Finally we show that compact operators have many invariant subspaces, leading to a triangular form and a theorem of Ringrose.