Let \(A\) be a Noetherian local ring and \(M\) a finitely generated \(A\)-module. In the case where \(\text{pd}_A M < \infty\), the structure of \(M\) is very rigid but we know a little bit if \(\text{pd}_A M = \infty\). In the present paper, the authors define a new invariant of \(M\), called the complete intersection dimension \(\text{CI-dim}_A M\), and investigate modules of finite complete intersection dimension. If \(\text{pd}_A M < \infty\), then \(\text{CI-dim}_A M = \text{pd}_A M\). However there are many modules of finite complete intersection dimension but infinite projective dimension. Therefore the notion of complete intersection dimension is meaningful in wide context. The authors give applications of complete intersection dimension: characterization of complete intersection; computing depth; asymptotic behavior of Betti numbers; minimality of Tate resolution and so on.