Flat band physics is a central theme in modern condensed matter physics. By constructing a system with vanishing momentum dispersion, it is possible to study physics in strongly interacting regimes. I will present work on two types of lattices that support flat bands: line graph lattices and glued tree lattices.  First, we study the properties of gapped flat bands in line graph lattices and use them as a guide to finding real materials with flat bands.  Additionally, we define and analyze several infinite families of lattices of glued trees in any number of spatial dimensions (and even some in hyperbolic space) that have only flat bands. The models that we introduce also have only compact localized states, despite their translation symmetries.