Fluid flow in porous systems driven by capillary pressure is one of the most ubiquitous phenomena in nature and industry, including petroleum and hydraulic engineering as well as material and life sciences. The classical Lucas-Washburn (LW) equation and its modified forms were developed and have been applied extensively to elucidate the fundamental mechanisms underlying the basic statics and dynamics of the capillary-driven flow in porous systems. The LW equation assumes that fluids are incompressible Newton ones and that capillary channels all have the same radii. This kind of hypothesis is not true for many natural situations, however, where porous systems comprise complicated pore and capillary channel structures at microscales. The LW equation therefore often leads to inaccurate capillary imbibition predictions in such situations. Numerous studies have been conducted in recent years to develop and assess the modifications and extensions of the LW equation in various porous systems. Significant progresses in computational techniques have also been attained to further improve our understanding of imbibition dynamics. A state-of-the-art review is therefore needed to summarize the recent significant models and numerical simulation techniques as well as to discuss key ongoing research topics arising from various new engineering practices. The theoretical basis of the LW equation is first introduced in this review and recent progress in mathematical models is then summarized to demonstrate the modifications and extensions of this equation to various microchannels and porous media. These include capillary tubes with nonuniform and noncircular cross sections, discrete fractures, and capillary tubes that are not straight as well as heterogeneous porous media. Numerical studies on the LW equation are also reviewed, and comments on future works and research directions for LW-based capillary-driven flows in porous systems are listed.