The synthetic dimension, a research topic of both fundamental significance and practical applications, has been attracting increasing attention in recent years. In this paper, we propose a theoretical framework to construct arbitrary synthetic dimensions, or N-boson synthetic lattices, using multiple bosons on one-dimensional lattices. We show that a one-dimensional lattice hosting N indistinguishable bosons can be mapped to a single boson on an N-dimensional lattice with high symmetry. Band structure analyses on this N-dimensional lattice can then be mathematically performed to predict the existence of exotic eigenstates and the motion of N-boson wave packets. As illustrative examples, we demonstrate the edge states in two-boson Su-Schrieffer-Heeger synthetic lattices without interactions, interface states in two-boson Su-Schrieffer-Heeger synthetic lattices with interactions, and weakly bound triplon states in three-boson tight-binding synthetic lattices with interactions. The interface states and weakly bound triplon states have not been thoroughly understood in previous research. Our proposed theoretical framework hence provides an interesting perspective to explore the multiboson dynamics on lattices with boson-boson interactions, and opens up a future avenue in the field of multiboson manipulation in quantum engineering.