Abstract Let us start by considering the symmetric random walk, which in each time unit is equally likely to take a unit step either to the left or to the right. That is, it is a Markov chain with Pi,i+1=12=Pi,i-1,i=0,±1,…. Now suppose that we speed up this process by taking smaller and smaller steps in smaller and smaller time intervals. If we now go to the limit in the right manner what we obtain is Brownian motion. We study the Brownian motion process both when its distribution is symmetric about 0 and when it has a drift away from 0. In both cases we derive the distribution of the maximum of the process up to a specified time.