The authors study the asymptotic distribution of certain U-statistics whose entries are local summary statistics of an underlying weakly dependent process \({({X_t})_{t \in Z}}\). They consider local statistics \({W_{n,j}} = \sqrt {{l_n}} g\left( {\frac{1}{{{l_n}}}\sum_{t \in {B_{n,j}}} {{X_t}} ,\frac{1}{{{l_n}}}\sum_{t \in {B_{n,j}}} {X_t^2} ,\dots,\frac{1}{{{l_n}}}\sum_{t \in {B_{n,j}}} {X_t^m} } \right)\), \(1 \le j \le {b_n}\), where \({B_{n,j}}: = \{ (j - 1){l_n} + 1,\dots,j{l_n}\} \) and block length \({l_n}\) tend to infinity. They establish asymptotic normality of such U-statistics. After that they study U-statistics of the type \({U_n}: = \frac{1}{{{b_n}({b_n} - 1)}}\sum\limits_{1 \le j \ne k \le {b_n}} {h({W_{n,j}},{W_{n,k}})} \).