Summary: The class \textbf{No} of surreal numbers, which John Conway discovered while studying combinatorial games, possesses a rich numerical structure and shares many arithmetic and algebraic properties with the real numbers. Some work has also been done to develop analysis on \textbf{No}. In this paper, we extend this work with a treatment of functions, limits, derivatives, power series and integrals. We propose surreal definitions of the arctangent and logarithm functions using truncations of Maclaurin series. Using a new representation of surreals, we present a formula for the limit of a sequence, and we use this formula to provide a complete characterization of convergent sequences and to evaluate certain series and infinite Riemann sums via extrapolation. A similar formula allows us to evaluate limits (and hence derivatives) of functions. By defining a new topology on \textbf{No}, we obtain the intermediate value theorem even though \textbf{No} is not Cauchy complete, and we prove that the fundamental theorem of calculus would hold for surreals if a consistent definition of integration exists.