Publisher Summary This chapter discusses the description of the different formulations of the general Daniell scheme of extending certain linear functional on a vector space. A brief discussion of the Daniell extension procedure is provided in the chapter. The extended function class of summable functions is defined through three classic methods, which include equality of the upper and lower integrals, difference of monotone limits with boundedness in integral, and closure of the class of elementary functions. The Riemann upper and lower integrals are replaced by slightly different ones, called the Daniell upper and lower integrals. A variant functional approach to integration of interest considers the integral as an extension by continuity under a suitable seminorm is introduced in the chapter. The systematic search for more general basic systems or certain particular systems to obtain new applications as well as to unify methods has given special functional approaches to integration. It is suggested that nonadditive measures and integrals might be of interest to pursue with respect to future applications in problems that cannot be treated with additivity.