  Convolution semigroups of states on a quantum group form the natural
noncommutative analogue of convolution semigroups of probability measures on a
locally compact group. Here we initiate a theory of weakly continuous
convolution semigroups of functionals on a C*-bialgebra, the noncommutative
counterpart of locally compact semigroup. On locally compact quantum groups we
obtain a bijective correspondence between such convolution semigroups and a
class of C_0-semigroups of maps which we characterise. On C*-bialgebras of
discrete type we show that all weakly continuous convolution semigroups of
states are automatically norm-continuous. As an application we deduce a known
characterisation of continuous conditionally positive-definite Hermitian
functions on a compact group.
