"Despite over a century of interest, the function of zebra stripes has never been examined systematically."

The quote is from an extensive 2014 study published in Nature Communications. The study examined the five prominent theories regarding the zebra stripes question. They are:

• 1) as a form of visual crypsis (camouflage) - probably matching a woodland background
  
• 2) disrupting predatory attack by visual-dazzling effects
  
• 3) reducing thermal load
  
• 4) having a social function.
  
• 5) avoiding ectoparasite attack
  

The 2014 research team excluded :

Hypotheses 1 & 2) "Despite the popularity of various sorts of confusion hypotheses, our data provide little support for this idea."

Hypothesis 3) "Heat management does not appear as a driver for equid stripes"

Hypothesis 4) "A social function cannot explain striping in equids either"

Leaving hypothesis 5) which they suggested "lend(s) strong ecological support for striping being an adaptation to avoid biting flies."

However, a previous (2002) study (in Mammal Review, Volume 32, Issue 4) had pointed out that the fly-avoidance idea was "[…] the only hypothesis that has been tested experimentally, and the results of these tests are inconclusive."
And further, the Mountain Zebra which has the most prominent and defined stripes, does not live in areas prone to biting insects.
Another experimental study (in: Proceedings of the Royal Society, B) performed in 2009 (with horses) showed that the most advantageous colour for insect-avoidance was white only.
A 2019 study found that horseflies preferred dark colours over stripes.

(Note: There are many other, less mainstream ideas, for example another theory(in: Annales Zoologici Fennici Vol. 44, No.5) was put forward in 2007 - that the stripes might operate as "an amplifier of the individual's escape potential.")

Further reading:

'Zebra Stripes' by Tim Caro, University of Chicago Press, 2016

How the zebra got its stripes: a problem with too many solutions Royal Society Open Science, 2015