Nonlinear site response analyses can provide more accurate predictions of ground motions during an earthquake than equivalent linear procedures, especially for higher amplitude ground motions. However, the advantage of performing the nonlinear analysis is limited by the availability and accuracy of the models used to represent the soil profile. The availability of models that represent cemented soil is very limited because the response of such soil to dynamic loading is not well understood, even though natural cementation occurs in many seismically active areas. This research investigates the shear modulus and damping of cemented sand and cemented clay through laboratory tests with pulse generating transducers, bender elements and resonant column devices. The effect of different factors including the cement type, cement content, void ratio, confining pressure, sand particle properties and clay type have on the reduction of shear modulus and increase in damping with increasing shear strain as well as the maximum shear modulus are examined through the test results. New shear stiffness models, Simsoil-CS for cemented sand and Simsoil-CC for cemented clay, based on the model Simsoil by Pestana and Salvati, 2006 are proposed to represent dynamic properties such as maximum shear modulus, modulus and damping variation with the shear strain of the cemented sand and cemented clay. Model Simsoil-CS and Simsoil-CC were developed and evaluated with the data from this and previous studies. Model parameters are determined for different types of soil and cementation agents. The relationship between the unconfined compressive strength and the model parameter of the cement type and cement content are also investigated and presented for cemented sand and cemented clay respectively. That relationship can be used to estimate the cement type and cement content with a simple unconfined compressive test when those properties are unknown or difficult to determine. This research advances the understanding of cemented sand and cemented clay by providing a database of test results, and creation of models which can be used to predict the response of cemented soils to dynamic loads. This can be useful for performing earthquake site response analysis for naturally cemented sites or sites that have been improved by cementation.