Lattice translation is a fundamental symmetry in many condensed matter systems. In this talk I will examine the interplay of translation symmetry with fractionalized excitations that can emerge in topological phases. In the first part of the talk, I will discuss recent advances in Lieb-Schultz-Mattis (LSM) type theorems, which put microscopic constraints on the low-energy dynamics of translation-invariant lattice systems. I will discuss the interpretation of such constraints as manifestation of quantum anomalies and introduce new types of constraints in fermionic systems. In the second part, I will discuss the relation between mobility of quasiparticles and their transformation under translation symmetry. With this point of view I will provide a systematic classification of fractonic excitations in three-dimensional U(1) spin liquids, possibly enforced by global symmetries, and their realizations in a gauged coupled layer construction.