Extreme value statistics of Brownian trajectories in one dimension will be discussed. The maximum is defined as the largest position to date, and the maxima of two particles undergoing independent Brownian motion are compared. The probability that the two maxima remain ordered is inversely proportional to the one-fourth power of time. I also will discuss a few generalizations to unequal diffusion constants, larger number of particles, and higher spatial dimensions. Related recent results on first-passage problems involving extreme values will be briefly mentioned.