This Research Note presents the K-theory and KK-theory for real C*-algebras and shows that these can be successfully applied to solve some topological problems which are not accessible to the tools developed in the complex setting alone.

Introduction
1. K-theory for real C*-algebras

1.1. Real C*-algebras
1.2. Clifford algebras
1.3. Crossed products
1.4. Elementary K-theory for real C*-algebras
1.5. K-groups of special crossed products
1.6. Non-stable K-theory

2. Fredholm operators and KK-theory

2.1. Real Hilbert modules
2.2. Fredholm operators and classifying spaces
2.3. Kasparov's KK-theory
2.4. The intersection product
2.5. The periodicity theorem in KK-theory
2.6. KK-groups of special crossed products

3. Index theory for real operators

3.1. Real Spin{-63}superscript c -bundles and KR-orientation
3.2. Families of real pseudodifferential operators
3.3. The index theorem for real elliptic pseudodifferential operators
3.4. Foliations and C*-algebras
3.5. The real longitudinal index theorem
3.6. Applications

References
Index

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