Hacon and McKernan have proved that there exist integers such that if is a smooth variety of general type and dimension , then the pluricanonical maps are birational for . These values are typically very large: for example and . In this talk we will show that the -th canonical maps of smooth threefolds and fourfolds of general type have birationally bounded fibers for and respectively. Furthermore, we will generalize these results to higher dimensions in terms of the constants and we will discuss recent progress on a conjecture of Chen and Jiang.