Informally, a real number is normal in base 10 if each of the digits 0, 1, . . . , 9 shows up with frequency 1/10 in its decimal expansion, each pair of digits 00, 01, . . . , 99 shows up with frequency 1/100 in its decimal expansion and so on. 

There are many easily stated (but very difficult!) open questions revolving around normality. For example, even determining the normality of $\pi$ in any base appears to be far out of reach of modern mathematics. 

Depending on the interest of the audience, we may explore connections between normal numbers and other areas of math such as ergodic theory, descriptive set theory, computability theory, probability theory, and others.