There's a few things wrong with JPEG that are covered above and below, but the truth of the matter is this: practically speaking, it's plenty good enough and plenty ubiquitous enough to secure it's place for the foreseeable future.
Text and logos are already handled better by PNG and SVG. But for photos? A high quality JPEG will look perfect to nearly everyone, and just about nobody cares if they can save 15% on a still image when they're slinging tons more data watching HD videos. Especially when that file size saving comes at the expense of guaranteed support on every platform, application, and device made in the modern computing age.
At least as an end-user format, JPEG won't be replaced anytime in the near future regardless of it's fairly minor deficiencies.