We present results from an experimental investigation of the surface dynamics controlling morphology during CH@sub 4@/H@sub 2@ plasma and wet etching of GaAs (001) surfaces. In the case of plasma etching, height-difference correlation functions (extracted from AFM data) indicate that morphology development is dominated by diffusion below ~ 100 nm, while desorption/etching effects dominate on larger length scales. The crossover between these two regimes is temperature dependent. Below 600 K, oscillatory structure in the height correlation function, associated with a "rippled" surface morphology, arises due to an instability in the etching dynamics. We attribute this instability to Ehrlich- Schwoebel step-edge barriers. The behavior of the height-difference correlation function and the time-dependent roughening is found to be in good agreement with studies of continuum models based on the Kuramoto-Sivashinsky (KS) equation. However, the Laplacian term in the KS equation has components arising from both the ES barrier and etching/desorption effects so that the effect of the ES barrier in producing the instability is weakened, compared to thin film growth examples. Using theoretical expressions for the kinetic coefficients in the KS equation, we find that the Ehrlich-Schwoebel barrier is ~ 0.05 eV in this system. We will also present initial results studying "facetting" during wet etching of GaAs (001). The initial formation of anisotropic "hillock" structures develops in time to form a hill-and-valley structure along the (110) direction, with faces close to [111]. The length scale coarsens in time and is dependent on the etchant composition. Comparisons with theoretical studies of kinetically controlled crystal growth and etching will be discussed.