  We give an introduction to the theory of weak Hopf algebras proposed recently
as a coassociative alternative of weak quasi-Hopf algebras. We follow an
axiomatic approach keeping as close as possible to the "classical" theory of
Hopf algebras. The emphasis is put on the new structure related to the presence
of canonical subalgebras A^L and A^R in any weak Hopf algebra A that play the
role of non-commutative numbers in many respects. A theory of integrals is
developed in which we show how the algebraic properties of A, such as the
Frobenius property, or semisimplicity, or innerness of the square of the
antipode, are related to the existence of non-degenerate, normalized, or Haar
integrals. In case of C^*-weak Hopf algebras we prove the existence of a unique
Haar measure h in A and of a canonical grouplike element g in A implementing
the square of the antipode and factorizing into left and right algebra
elements. Further discussion of the C^*-case will be presented in Part II.
