After reviewing the Green-Schwarz superstring using the approach of Siegel, the super-string is covariantly quantized by constructing a BRST operator from the fermionic constraints and a bosonic pure spinor ghost variable. Physical massless vertex operators are constructed and, for the first time, N-point tree amplitudes are computed in a manifestly ten-dimensional super-Poincare´ covariant manner. Quantization can be generalized to curved supergravity backgrounds and the vertex operator for fluctuations aroundAdS5×S5is explicitly constructed. ©2002 American Institute of Physics.