Computational operations always rely on real physical processes, which are data input, data representation in a memory, data manipulating using algorithms and finally, the data output. With conventional computers all the processes are classical processes and can be described accordingly. Theoretically, it is known for several years now that certain computations could be processed much more efficiently using quantum mechanical operations. Therefore, there it would be desirable to built a quantum computer. This requires the implementation of quantum bits (qubits), quantum registers and quantum gates and the development of quantum algorithms. In this talk, several techniques for the realization of a quantum computer will be presented and compared. Special emphasis will be given to the ion storage and laser cooling techniques which are currently investigated for an application with quantum computers. First experimental results with trapped Ca+ ions in linear traps will be presented and discussed.