In this paper we briefly revisit Motzkin numbers before noting a novel manifestation of Motzkin numbers among certain fault-free tableaux. We next describe two other instances involving tableaux in which Motzkin numbers arise and present a bijection between them, yielding a new proof of the fact that standard Young tableaux with three or fewer rows are enumerated by the Motzkin numbers. We then extend the method of bijection to prove a new result that generalizes the original observation to larger tableaux. This approach motivates a natural way to generalize the Motzkin numbers, namely, as the sequences that result from fixing the height of the grid and letting the width vary. Finally, we note a few properties of these numbers and state two conjectures about their asymptotics.