The problem of iterative roots is a weak version of embedding flows,which is a basis of dynamic interpolation.However, even for one-dimensional mappings,it is still difficult to discuss the non-monotonicity and smoothness of their iterative roots.In this paper,some new results are introduced,especially for the existence and construction of continuous iterative roots of strictly piecewise monotone and continuous mappings,and for smoothness of iterative roots about local and global cases.Finally,as a special class of strictly piecewise monotone and smooth mappings,polynomials are discussed and the conditions of existence with calculation methods of their iterative roots are given.