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Some new families of small complete caps in $PG(N,q)$, $q$ even, aredescribed. By using inductive arguments, the problem of the construction ofsmall complete caps in projective spaces of arbitrary dimensions is reduced tothe same problem in the plane. The caps constructed in this paper provide animprovement on the currently known upper bounds on the size of the smallestcomplete 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 $q=2^{Cm},$ $C\geq 5,$ $m\geq9,$ with $C,m$ odd; and for all $q\leq 2^{18}.$

New inductive constructions of complete caps in $PG(N,q)$, $q$ even