  We describe some forms with greater Waring rank than previous examples. In
$3$ variables we give forms of odd degree with strictly greater rank than the
ranks of monomials, the previously highest known rank. This narrows the
possible range of values of the maximum Waring rank of forms in $3$ variables.
In $4$ variables we give forms of odd degree with strictly greater than generic
rank. In degrees greater than or equal to $5$ these are the first examples
showing that there exist forms with Waring rank strictly greater than the
generic value.
