The topological entropy of a nonautonomous dynamical system given by a sequence of compact metric spaces ${\left(X_i\right)}_{i=1}^{\infty }$ and a sequence of continuous maps ${\left(f_i\right)}_{i=1}^{\infty }$, $f_i:X_i\to X_{i+1}$, is defined. If all the spaces are compact real intervals and all the maps are piecewise monotone then, under some additional assumptions, a formula for the entropy of the system is obtained in terms of the number of pieces of monotonicity of $f_n○...○f_2○f_1$. As an application we construct a large class of smooth triangular maps of the square of type $2^{\infty }$ and positive...