We present approximation algorithms for the following problems: the two-dimensional bin packing (2BP), the three-dimensional packing problem (TPP) and the container packing problem (3BP). We consider the special case in which the items to be packed are small and must be packed on-line. We say an item is small if each of its dimension is at most 1/m of the respective dimension of the recipient, where m is an integer greater than 1. These problems are denoted by 2BP_m, TPP_m and 3BP_m, and the performance of the algorithms are given in terms of the parameter m. To our knowledge, the parametric on-line algorithms we present here have the best so far achieved asymptotic performance bounds for these problems. For 2BP_m and TPP_m we present algorithms with performance bound close to (m+2)/m+1/(m+1)², and for 3BP_m we describe an algorithm with performance bound close to (m+3)/m+2/m²+1/(m+1)². For m=2 (respectively m=3) these bounds are 2.112 and 3.112 (respectively 1.73 and 2.285). The results on 2BP_m and 3BP_m extend the results on multidimensional on-line packing presented by Coppersmith and Raghavan [Oper. Res. Lett. 8(1) (1989) 17]. The result on TPP_m improves the bound (m+1)/(m−1) due to Li and Cheng [SIAM J. Comput. 19 (1990) 847]. All these algorithms can be implemented to run in O(nlogn) time, where n is the number of items to be packed.