Update 2020

As of version 1.8.5, fontspec is now supported by the local typesetter (hyperref too).

Update June 30, 2016

YIKES! I spoke to soon. With such anticipation, I jumped the gun and realized that my document with fontspec was typesetting with the cloud typesetter. Still waiting to say: "Texpad's local (on-device compiler) and cloud compiler both support fontspec and work great!" Thanks to cfr and Joseph Wright for helping me notice this. Only the cloud compiler supports fontspec.

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The developers claim to provide fontspec support starting with version 1.7. I have been waiting over 3 years for this.

#Update April 2, 2016

Version 1.7 is currently in beta testing. We hope to release it in 2-3 weeks. It includes a hugely improved typesetter with beamer, tikz and a host of other issues resolved. Fontspec (or support for local fonts in general) will follow later this summer.

The app also delivers new UI that matches Texpad OS X and hence is a lot more productive on larger iPads.

Fontspec

XeTeX and LuaTeX both support unicode and TTF fonts natively through the fontspec package. This is in testing right now, and it will be available on both platforms from 1.7.

#Source https://www.texpadapp.com/support/ios/typesetting/vvtex-support

#Release Notes (as of this answer)

The past few updates, but still not v.1.7 from https://www.texpadapp.com/releasenotes/tpios:

Version 1.6.20
Misc. dropbox file access fixes
Version 1.6.19
Misc. file handling improvements
Backup manager for edited documents
64 bit support (updated OpenSSL)
Version 1.6.18
Dropbox error handling improved.
Dropbox move/copy bug fixed where multiple move/copy calls would occasionally fail.
Version 1.6.17
A fix for Dropbox uploads for cloud typeset PDFs