Flux-Corrected Transport has proven to be an accurate and easy to use an algorithm to solve nonlinear, time-dependent continuity equations of the type which occur in fluid dynamics, reactive, multiphase, and elastic plastic flows, plasmadynamics, and magnetohydrodynamics. This report updates and supersedes a previous report entitled Flux-Corrected Transport Modules for Solving Generalized Continuity Equations. It can be used as a user manual for subroutines and test programs included in the appendices. The entire LCPFCT library in its most recent from is presented and discussed in detail. There are, in addition, discussions of more general topics such as the application of physical boundary conditions, physical positivity and numerical diffusion which help to put the numerical aspects to this subroutine library in context.