We present a combinatorial explanation of $(\alpha,\beta,\gamma)$-Motzkin and Riordan paths by using Riordan arrays. In addition, we prove some well-known combinatorial identities by algebraic methods. Futhermore, we apply the symbolic method or the object grammars and Lagrange inversion formula to Motzkin paths with colored ascents to get some insteresting combinatorial identities, in which the ascents colored by Dyck path, Motzkin path, Schr\"oder path, small Schr\"oder path, Catalan rook path and Delannoy path, respectively.