Having read many books on the Byzantine Empire in all stripes, I can hands down absolutely recommend "The Making of Orthodox Byzantium: 600-1025" by Mark Whittow as the best straddle between accurate and readable.

It answers some of the most common questions, as well as gives alternate looks at some well known myths. The style is very concise and straightforward, and doesn't overwhelm the reader. I read the book over the course of three days, to give you an indication of its ease. It is also, quite universally recognized by other scholars, as one of the best general overviews of that period.

A few notes on the books suggested by the others.

Ostrogorsky is still considered a solid book, though parts of it are out of date. However it can be a heavy read and sometimes the quirks of translation from another native language come out in the style.

Treadgold is very comprehensive and readable, however you should consider him a bit of a Byzantine "maximalist" (in comparison to Whittow who's more of a "minimalist") as Treadgold tends to portray Byzantium as equal to the old Roman Empire in military power and efficiency. He's more of an old school political historian than some of the socio-economic historians that are in vogue today. So read him with a wary eye, knowing that he represents the upper (though still accepted) boundaries of the scholarship.

http://www.amazon.com/A-History-Byzantine-State-Society/dp/0804726302

I know it's a bit pricey, and I haven't read the book, but I have seen lectures by the author, Warren Treadgold, from St. Louis University. He's very intelligent, and this is a very popular book, it seems, by far his most popular. He's very qualified in terms of Byzantine history, in fact it's almost his entire career! The book itself is from 1997, so I'm not sure you'd call it "current," per se, but I'm not positive much information has surfaced in the past 18 years that changes anything he would say in this book. As you can see on that link though, he has many other books published about Byzantium and the Byzantine Empire both before and after his 1997 book, if you're interested in that.

I'd also suggest A History of the Byzantine State by the Yugoslavian Byzantinist George Ostrogorsky. While older than Treadgold's work(which is also very good), it's cheaper and shorter yet is still considered a definitive history--I read Ostrogorsky's during multiple Byzantine history courses and found the writing style very accessible and not overly concerned with minutae while still providing lots of detail.

Sailing from Byzantium by Colin Wells is a reasonable popular history, and a pretty solid introduction to the general environment and overview of the empire.