Brownian motion and its role in financial models

Describe Brownian motion and explain how it is used in quantitative finance.

A continuous random walk with independent Gaussian increments. Brownian motion starts at zero, has continuous but nowhere differentiable paths, and increments over $[s,t]$ distributed $N(0,t-s)$, so uncertainty grows like $\sqrt t$. Finance uses it as the source of randomness in SDEs: GBM for stock prices, mean-reverting models for rates, and as the basis for Ito calculus, Black-Scholes and Monte Carlo simulation. Its Markov and martingale properties make models tractable, though real returns have fatter tails and jumps.