  We introduce the concepts of Grover operators and Grover kernels to
systematically analyse Grover's searching algorithms. Then, we investigate a
one-parameter family of quantum searching algorithms of Grover's type and we
show that the standard Grover's algorithm is a distinguished member of this
family. We show that all the algorithms of this class solve the searching
problem with an efficiency of order $O(\sqrt{N})$, with a coefficient which is
class-dependent. The analysis of this dependence is a test of the stability and
robustness of the algorithms. We show the stability of this constructions under
perturbations of the initial conditions and extend them upon a very general
class of Grover operators.
