  Some new families of small complete caps in $PG(N,q)$, $q$ even, are
described. By using inductive arguments, the problem of the construction of
small complete caps in projective spaces of arbitrary dimensions is reduced to
the same problem in the plane. The caps constructed in this paper provide an
improvement on the currently known upper bounds on the size of the smallest
complete cap in $PG(N,q),$ $N\geq 4,$ for all $q\geq 2^{3}.$ In particular,
substantial improvements are obtained for infinite values of $q$ square,
including $ q=2^{2Cm},$ $C\geq 5,$ $m\geq 3;$ for $q=2^{Cm},$ $C\geq 5,$ $m\geq
9,$ with $C,m$ odd; and for all $q\leq 2^{18}.$
