Interpreting the noise in a stochastic differential equation, in particular the Itô versus Stratonovich dilemma, is a problem that has generated a lot of debate in the physical literature. In the last decades, a third interpretation of noise, given by the so-called Hänggi-Klimontovich integral, has been proposed as better adapted to describe certain physical systems, particularly in statistical mechanics. Herein, we introduce this integral in a precise mathematical manner and analyze its properties, signaling those that have made it appealing within the realm of physics. Subsequently, we employ this integral to model some statistical mechanical systems, such as the random dispersal of Langevin particles and the relativistic Brownian motion. We show that, for these classical examples, the Hänggi-Klimontovich integral is worse adapted than the Itô integral and even the Stratonovich one.