We study several enumerative properties of the set of all binary strings without zigzags, i.e., without substrings equal to 101 or 010 . Specifically we give the generating series, a recurrence and two explicit formulas for the number wm,n of these strings with m 1’s and n 0’s and in particular for the numbers wn = wn,n of central strings. We also consider two matrices generated by the numbers wm,n and we prove that one is a Riordan matrix and the other one has a decomposition LTL where L is a lower triangular matrix and T is a tridiagonal matrix, both with integer entries. Finally, we give a combinatorial interpretation of the strings under consideration as binomial lattice paths without zigzags. Then we consider the more general case of Motzkin, Catalan, and trinomial paths without zigzags.