Lattice translation is a fundamental crystalline symmetry in many condensed matter systems.In this talk we examine the interplay of translation symmetry with topological order. In the first part of the talk, we will discuss recent advances in Lieb-Schultz-Mattis (LSM) type theorems, which put stringent microscopic constraints on the low-energy dynamics of translation-invariant lattice systems. We will introduce a new interpretation of the classic LSM theorem and its higher-dimensional version by Oshikawa and Hastings, as consequences of the bulk-boundary correspondence for translation symmetry-protected topological phases in higher dimensions. We then discuss various generalizations and refinements following this new perspective. In the second part, we discuss the relation between translation symmetry transformations on quasiparticle excitations and their mobility. It turns out that certain patterns of translation symmetry fractionalization (in the presence of a global U(1) symmetry) imply that quasiparticles are immobile, i.e. they become fractons. We show that such fractonic particles naturally appear in three dimensional U(1) spin liquids, and develop a systematic understanding for symmetry enforced restrictions on mobility. We will also discuss possible physical realizations.