We investigate some problems for truncated Toeplitz operators. Namely, the solvability of the Riccati operator equation on the set of all truncated Toeplitz operators on the model space K (theta) = H (2)Iy theta H (2) is studied. We study in terms of Berezin symbols invertibility of model operators. We also prove some results for the Berezin number of the truncated Toeplitz operators. Moreover, we study some property for H (2)-functions in terms of noncyclicity of co-analytic Toeplitz operators and hypercyclicity of model operators.