hacon and mckernan have proved that there exist integers $r_n$ such that if $x$ is a smooth variety of general type and dimension $n$, then the pluricanonical maps $|rk_x|$ are birational for all $r\geq r_n$. these values are typically very large: for example $r_3\geq 27$ and $r_4\geq 94$. in this talk we will show that the $r^{\textup{th}}$ canonical maps of smooth threefolds and fourfolds of general type have birationally bounded fibers for $r\geq 2$ and $r\geq 4$ respectively. furthermore, we will generalize these results to higher dimensions in terms of the constants $r_n$ and we will discuss recent progress on a conjecture of chen and jiang.