K-theory for Cuntz–Krieger algebras is very important not only for classification theory of \(C^*\) -algebras but also for classification theory for topological Markov shifts. In this chapter, we will first introduce K-theory for general \(C^*\) -algebras in brief. We will second study infinite projections in unital simple \(C^*\) -algebras. We will third describe definition of purely infinite \(C^*\) -algebra and study characterization of the purely infiniteness of unital simple \(C^*\) -algebras. We will finally give a brief introduction to the \({{\operatorname {Ext}}}\) -groups for \(C^*\) -algebras.