Hi,
Earliest models of MOSFET devices were SPICE Level 1, Level 2 and Level 3. They were primitive models before more advanced BSIM models were developed. BSIM models were developed in UC Berkeley. BSIM 3v3 model was one of the successful models that is used widely in analog design community. Now, there is BSIM6 which is the most advanced model in this line. Advanced models are more complex than primitive models. They include more parameters and they can model more features of MOS transistors. For example, primitive models have very basic flicker noise model for MOS transistors. But, BSIM 4 and BSIM 6 have more accurate flicker noise model of MOS transistors.
Apart from BSIM line, there are also other MOS models. PSP model (developed by NXP Semiconductors & Arizona State University) and EKV model (developed in EPFL) are the other succesful models. Each of them have pros and cons regarding accuracy, numerical stability and complexity.