This paper presents and explores a diffusion model that generalizes Brownian Motion (BM). On the one hand, as BM: the model's mean square displacement grows linearly in time, and the model is Gaussian and selfsimilar (with Hurst exponent $\frac{1}{2}$). On the other hand, in sharp contrast to BM: the model is not Markov, its increments are not stationary, and its non-overlapping increments are not independent. Moreover, the model exhibits a host of statistical properties that are dramatically different than those of BM: aging and anti-aging, positive and negative momenta, correlated velocities, persistence and anti-persistence, aging Wiener-Khinchin spectra, and more. Conventionally, researchers resort to anomalous-diffusion models -- e.g. fractional BM and scaled BM (both with Hurst exponents different than $\frac{1}{2}$) -- to attain such properties. This model establishes that such properties are attainable well within the realm of diffusion. As it is seemingly Brownian yet highly non-Brownian, the model is termed \emph{Weird Brownian Motion}.