  We define non-commutative schemes by using prime ideals of non-commutative
rings, and discuss the \'etale cohomology, the Betti cohomology, and the
fundamental groups of non-commutative schemes. For non-commutative schemes
which are finite over centers, we prove the finiteness theorem for the higher
direct images in \'etale cohomology theory, and the comparison theorem between
\'etale cohomology and Betti cohomology. In Appendix, for non-commutative
schemes over finite fields which are finite over centers and satisfy a certain
condition, $L$-functions are expressed by using \'etale cohomology with compact
supports.
