Conditions for integrability Michael Spivak, in his "Calculus" writes 

Although it is possible to say precisely which functions are integrable,the criterion for integrability is too difficult to be stated here

I request someone to please state that condition.Thank you very much!

Just a small clarification (as an answer as I cannot comment):
The way in which mixedmath wrote the answer might lead to confusions, a different way would be:
A bounded function $f$ on $[a,b]$ is Riemann integrable iff it is continuous almost everywhere.
Note that this assumes the function to be bounded and it is not an implication of the theorem (For instance think of $f=\frac{1}{\sqrt{(x)}}$, whose integral in $[0,1]$ is 2 but it is not bounded).
