Apologies for any factual/mathematical innacuracies in what follows... I am not a mathematician or cryptologist, though I do believe I have a basic understanding...
Encryption was developed for the purposes of protecting communications from eavesdroppers. From the earlest letter-shifting techniques, through Germany's Enigma machines in WW2, and beyond to current RSA methods, all have served their purposes well. At least until each, in turn, was worthwhile dedicating some serious mathematical brainpower (as in Enigma) to defeating it.
Per communications data I find current technologies very convincing indeed, to the extent I trust them completely. As far as I understand the technology, the maths of prime factorials, while unproven, are widely acceptable as being good, and keys are machine generated and evaporate after the communications is complete, so nobody's toenails to pull.
But the real distinction that worries me is nowadays, people think encryption can be used not just to protect communications, but also to protect stored data. That is my criticism, based on all historical precedent they are wrong. If if the data is that sensitive then I would say either...
A) don't store it
or
B) ensure that nothing bad will happen if it escapes
or
C) accept that if it escapes it is your own fault, the fact it was encrypted is no defence at all.
I have not the slightest doubt the day will come when somebody finds a mathematical flaw in prime factorials and RSA. It will come as a shock, but no more of a shock that cracking Enigma was to 1930s Germany. But when that day comes, I shall feel vindicated that I have no online bank accounts.