I will survey our recent results on the joint distributions and correlations of
the maxima m and M achieved by an unconstrained Brownian motion on the time intervals [0,t_1] and [0,t_2] with t_2 > t_1. In particular, I will show how to evaluate exact forms of the distribution functions P(m,M) and P(G=M-m), and calculate the moments E(M -m)^k and the cross-moments E(m^l)(M^k) with arbitrary integers l and k. Analogous results for Brownian Bridges will also be presented. As an application of the obtained joint distributions, I will discuss a possibility of extracting the ensemble-average diffusion coefficient in single-trajectory experiments using a single realization of the maximum process of a Brownian motion.