Partial sums and sample means of r-dimensionally indexed arrays of independent random variables have been studied by Dunford (1951), Zygmund (1951), Kuelbs (1968), Wichura (1969), Smythe (1973), Gut (1978, 1992), Etemadi (1981), Klesov (1981, 1983), and Su and Taylor (1992), whose results cover weak convergence, invariance principles, and almost sure behavior. Applications of such results arise in ergodic theory and the study of Brownian sheets. This paper extends to the case of U-statistics, for example, the sample variance, defined on such an array. An almost sure representation as an i.i.d. average, the central limit theorem for the case of random index, the law of the iterated logarithm, and an invariance principle are developed.