Side channel collision attacks are a powerful method to exploit side channel leakage. Otherwise than a few exceptions, collision attacks usually combine leakage from distinct points in time, making them inherently bivariate. This work introduces the notion of near collisions to exploit the fact that values depending on the same sub-key can have similar while not identical leakage. We show how such knowledge can be exploited to mount a key recovery attack. The presented approach has several desirable features when compared to other state-of-the-art collision attacks: Near collision attacks are truly univariate. They have low requirements on the leakage functions, since they work well for leakages that are linear in the bits of the targeted intermediate state. They are applicable in the presence of masking countermeasures if there exist distinguishable leakages, as in the case of leakage squeezing. Results are backed up by a broad range of simulations for unprotected and masked implementations, as well as an analysis of the measurement set provided by DPA Contest v4.