  In this work we study algebraic, geometric and topological properties of the
Milnor classes of local complete intersections with arbitrary singularities. We
describe first the Milnor class of the intersection of a finite number of
hypersurfaces, under certain conditions of transversality, in terms of the
Milnor classes of the hypersurfaces. Using this description we obtain a
Parusi\'{n}ski-Pragacz type formula, an Aluffi type formula and a description
of the Milnor class of the local complete intersection in terms of the global
L\^e cycles of the hypersurfaces that define it. We consider next the general
case of a local complete intersection $Z(s)$ defined by a regular section $s$
of a rank $r$ holomorphic bundle $E$ over a compact manifold $M$, $r \geq 2$.
We notice that $s$ determines a hypersurface $Z(\tilde s)$ in the total space
of the projectivization $\mathbb{P}(E^{\vee})$ of the dual bundle $E^{\vee}$,
and we give a formula expressing the total Milnor class of the local complete
intersection $Z(s)$ in terms of the Milnor classes of the hypersurface
$Z(\tilde s)$.
