Originally discovered in condensed matter systems, topological insulators (TIs) have been ubiquitously extended to various fields of classical wave physics including photonics, phononics, acoustics, mechanics, and microwaves. In the bulk, like any other insulator, TIs prohibit energy propagation. On their surface, however, they support one-way propagative states with inherent protection against certain types of disorders and defects. In this work, I explore the possibility of performing advanced signal processing and analog computing tasks based on the boundary states of wave topological insulators. By providing numerical and experimental verifications, I demonstrate that such kind of computing scheme, referred to as topological analog signal processing, provides one with strong robustness against imperfection and disorder. It is in sharp contrast to conventional signal processors, which are often very sensitive to geometrical tolerances. Going a step further, I even demonstrate that, in some topological systems with specific parameter ranges, the harmful effects of the disorder can be turned into an asset so as trigger functionalities of interest. These findings open up large perspectives for a new generation of all-optical signal processors that are not only fast and power-efficient but also offer strong levels of reliability and robustness to disorder.