According to MPEG-LA's most recent license terms summary for end-users and non-profit organizations, the terms of the H.264 license will remain until December 21, 2020. Essentially, the relatively "lax" H.264 licensing standards will remain as-is for most non-industry users until 2021. For this reason, we can expect H.264 to remain a force in the digital video industry for at least a few more years. It's well-supported on almost all devices, playback scenarios, and browsers.
It should be noted that x264 is not the most efficient codec anymore in terms of the ratio of perceived visual quality : file size, but it is definitely the most hastle-free codec to use. If used properly, few users should have problems playing back the content held within an MP4 file made with x264.
That being said, keep an eye out for the Daala codec by the Xiph Foundation. They're the same people responsible for OPUS, Ogg, FLAC, Speex, and Theora. Followingly, Daala is a royalty-free codec, which is a huge deal because patent/royalty issues are among the most effective ways to give digital video folks like me elevated blood pressure. There are a few very intriguing concepts being developed for Daala, the details of which are available at the project homepage. All said, in a few years' time, Daala could be eating codecs like HEVC and VP10 for breakfast.