This paper considers nonparametric estimation of Brownian semi-stationary processes (BSS; also known as continuous-time moving-average processes). We establish fully nonparametric identification of the so-called kernel functions of the processes through their covariance functions. This identification result allows us to propose a nonparametric series/sieve estimator of the kernel functions. We also investigate asymptotic properties of the proposed estimator, as well as its finite-sample properties through a simulation study.