Abstract:  The Riemann integral is familiar to all calculus students.  However there are many shortcomings with this integral.  For example, it was recognized that the Riemann integral has poor convergence properties; for example, a function which is the pointwise limit of a uniformly bounded sequence of integrable functions need not be Riemann integrable. Also, the hypotheses in the fundamental theorem of calculus require that a derivative be Riemann integrable before we can say that the integral from a to x of F'(t) is F(x) - F(a).  There are two distinct integrals which are defined with a slight adjustment in the definition of the Riemann integral.  The McShane integral is equivalent to the Lebesgue integral, and overcomes the convergence problems of the Riemann integral.  The Henstock integral is more general than the Lebesgue integral, and has the most general form of the fundamental theorem of calculus.  All derivatives are integrable, which is not the case even for the Lebesgue integral.  I will describe each integral, along with the relevant properties that each integral has, without using any measure theory.