Let \(B\) be a real separable Banach space with the norm \(\|\cdot\|\). Let \(X_ 1,\dots, X_ n\) be independent random elements with values in a measurable space \(({\mathcal X},{\mathcal A})\) and having identical distribution \(P\). Each symmetric function \(\Phi: {\mathcal X}^ m\to B\) defines a so-called \(UB\)-statistic \[ U_{mn}= {n\choose m}^{-1} \sum_{1\leq i_ 1\leq\dots\leq i_ m\leq n} \Phi(X_{i_ 1}, \dots, X_{i_ m}). \] If \(B=H\), where \(H\) is a separable Hilbert space, then \(U_{mn}\) is referred to as \(UH\)-statistic. If \(E\| \Phi\| <\infty\), then \(U_{mn}\) is an unbiased estimate of the \(B\)-valued element \[ \Theta= \Theta(P)= E \Phi(X_ 1,\dots, X_ m). \] The martingale structure, estimates of moments, the law of large numbers, the central limit theorem, the invariance principle, estimates of the rate of convergence, and large deviations are established. The canonical Hoeffding representation of the statistic \(U_{mn}\) is the main point investigating the limit behaviour of \(UB\)-statistics.