There is a significant need for 3-D steady-state and transient neutron transport formulations and codes that yield accurate, high-fidelity solutions with reasonable computing resources and time. These tools are essential for modeling innovative nuclear systems, such as next-generation reactor designs. The existing methods generally compromise heavily between accuracy and affordability in terms of computation times. In this dissertation, novel algorithms for simulation of reactor transient conditions have been developed and implemented into the RAPID code system. In addition, extensive computational verification and experimental validation of RAPID's steady-state and transient algorithms was performed, and a novel virtual reality system (VRS) web-application was developed for the RAPID code system. The new algorithms, collectively referred to as tRAPID, are based on the Transient Fission Matrix (TFM) methodology. By decoupling the kinetic neutron transport problem into two different stages (an accurate pre-calculation to generate a database and an on-line solution of linear partial differential equations) the method ensures the preservation of the highest level of accuracy while also allowing for high-fidelity modeling and simulation of nuclear reactor kinetics in a short time with minimal computing resources. The tRAPID algorithms have been computationally verified using several computational benchmarks and experimentally validated using the JSI TRIGA Mark-II reactor. In order to develop these algorithms, first the steady-state capabilities of RAPID have been successfully benchmarked against the GBC-32 spent fuel cask system, also highlighting issues with the standard eigenvalue Monte Carlo calculations that our code is capable of overcoming. A novel methodology for accounting for the movement of control rods in the JSI TRIGA reactor has been developed. This methodology, referred to as FM-CRd, is capable of determining the neutron flux distribution changes due to the presence of control rod in real-time. The FM-CRd method has been validated with successfully using the JSI TRIGA reactor. The time-dependent, kinetic capabilities of the new tRAPID algorithm have been implemented based on the Transient Fission Matrix (TFM) method. tRAPID has been verified and validated using the Flattop-Pu benchmark and reference calculations and measurements using the JSI TRIGA reactor. In addition to the main tRAPID algorithms development and benchmarking efforts, a new web-application for the RAPID Code System for input preparation and interactive output visualization was developed. VRS-RAPID greatly enhances the usability, intuitiveness, and outreach possibilities of the RAPID Code System.