Freeform optics, broadly described as optical elements with at least one surface lacking translational or rotational symmetry, have seen increased interest due to recent advances in optical design, fabrication, and measurement. Freeform optics provide additional design freedoms and capabilities that enable greater control of wavefronts and new functionality compared to conventional optics. For these reasons, freeform optics are receiving great interest for imaging, metrology, illumination, and many other applications. However, most freeform elements are designed for monochromatic light sources, especially in the case of transmissive optics. Studies on the advantages, disadvantages, and applications of freeforms with broadband sources have been limited to some illumination systems and to reflective systems which are inherently achromatic. The primary goal of this dissertation is to characterize examples of transmissive freeform optics in applications where broadband optical functionality is not currently enabled or where freeform optics enable enhanced broadband functionality. Results are addressed through three articles. The first article presents a broadband variable transmission sphere based on freeform Alvarez lenses to facilitate Fizeau interferometry across a broad range of f-numbers and source wavelengths. The second article demonstrates the use of additional freeform design freedoms to reduce undesirable retrace errors in broadband variable transmission spheres. The third article reports on the design and performance of freeform transmissive elements for compact and efficient broadband laser beam shaping. These three articles illustrate advantages and challenges for the use of freeform optics in broadband optical systems.