Does Monero have any Android or iOS wallets available?

No Monero iOS or Android wallet currently existed in the Apple App Store or Google Play Store for a long time with the custodial Freewallet exception related to Bytecoin/Minergate. This wallet has been marked as distrusted by the Monero community in the Monero Reddit wiki:

Freewallet – Please read through the many threads out there regarding Freewallet, and the selective scam that it plays on its unsuspecting users

If there are no mobile wallets today, are there any in development?

Monero Research Lab researcher and developer Shen Noether, aka NobleSir created beta wallets for both Android and Windows 10 in 2016 but their respective GitHub pages were closed following his MRL departure. Jaxx spent some time working on integrating Monero (for both Android and iOS) but temporarily halted Monero development for technical reasons (development pre RingCT needed to be altered after the v4 fork in January 2017. Reportedly they have now resumed plans to integrate Monero.

Currently Android development is the furthest along for:

Coinomi is reportedly close to completion and in the testing phase

MyMonero is close to complete pending porting iOS code into the Android JavaScript App. Its source code can be found in GitHub.

Official GUI source code can be tested now, with no release dates known at this time

Monerujo is an open source app now available for beta testing in Google Play. Following more testing and bug fixes its developer also plans to add it to F-Droid.

Late stage iOS development has been reported for iOS by:

MyMonero is complete pending a DUNS number required for Apple App Store approval and a possible small delay to ensure simultaneous cross platform release

XWallet is waiting for iOS approval

What unique privacy or security concerns do Monero mobile wallets have?

Privacy concerns are present anytime you rely on the full node of a 3rd party instead of running your own. Some of the mobile wallets listed above allow users to connect to their own server.

Security risks are highest among closed source mobile applications and those which provide your private keys to a third party. Implementation related errors can occur in any application but are more common with apps that have not received sufficient testing, peer review or that require users to complete complicated setup procedure.