A wide variety of topological materials presenting intriguing states of quantum matter have mostly been discovered in two-dimensional and three-dimensional systems. Recently, a novel quasi-one-dimensional family of bismuth halogenides, Bi4X4 (X = I, Br), has attracted considerable attention because it can be easily tuned by external pressure and cleaved in practical applications. Here, based on our theoretical predictions and experimental observations, we give a brief progress report on the study of these materials which exhibit rich phase diagrams including strong/weak and high-order topological insulators, composite Weyl semimetals. Many novel properties of these materials are also reported, including a room-temperature quantum spin Hall edge state based scanning tunnelling microscopy, pressure-induced superconductive and structural phase transitions, strong infrared absorption of the topologically originated edge states as well as a substantially increased carrier lifetime for the boundary states. These findings provide critically valuable information to understand the physics behind the new topological phenomena and offer a proof-of-principal methodology for studying their practical applications in the field of topological electronics, optoelectronics and spintronics.