Publisher Summary This chapter promotes, details, and exploits the fact that splines—that is, smooth piecewise polynomial functions, are weighted sums of B-splines. It provides definitions of piecewise polynomials and B-splines. B-splines are defined in terms of a knot sequence. The chapter describes the properties of B-spline, such as recurrence relation, support and positivity, Marsden's identity, partition of unity, knot averages, local linear independence, dual functionals, good condition, convex hull, and differentiation. The chapter discusses polynomials and piecewise polynomials in the spline space.