A smooth projective variety is a type of geometric object studied in algebraic geometry. It is defined as a projective variety that is also smooth, meaning it has no singular points where the structure is not well-defined. These varieties can be thought of as higher-dimensional generalizations of curves and surfaces, and they are often described using polynomial equations. Projective varieties are embedded in projective space, which allows for a natural way to study their properties. The smoothness condition ensures that the variety behaves nicely, making it easier to apply tools from both algebra and geometry. Examples include complex projective spaces and elliptic curves.