Biological samples continuously emit ultra-weak photon emission (UPE, or "biophotons") which stems from electronic excited states generated chemically during oxidative metabolism and stress. Thus, UPE can potentially serve as a method for noninvasive diagnostics of oxidative processes or, if discovered, also of other processes capable of electron excitation. While the fundamental generating mechanisms of UPE are fairly elucidated together with their approximate ranges of intensities and spectra, statistical properties of UPE is still a highly challenging topic. Here we review claims about nontrivial statistical properties of UPE, such as coherence and squeezed states of light. After introduction to the necessary theory, we categorize the experimental works of all authors to those with solid, conventional interpretation and those with unconventional and even speculative interpretation. The conclusion of our review is twofold; while the phenomenon of UPE from biological systems can be considered experimentally well established, no reliable evidence for the coherence or nonclassicality of UPE was actually achieved up to now. Furthermore, we propose perspective avenues in the research of statistical properties of biological UPE.