The fractional quantum Hall effect is one of the most exotic collective phenomena discovered in nature that has triggered the ideas of emergent topological order, fractional statistics, and many other novel concepts. A powerful tool to study the fractional quantum Hall effect is the construction of microscopic trial wave functions, which not only capture the topological features of the physical states in the fractional quantum effect but also allow quantitative calculations of various observables that can be compared to experimental results. A broad class of fractional quantum Hall states is described by the composite fermion theory. Composite fermions are the emergent bound states of electrons and even number of quantized vortices. The microscopic wave functions of composite fermions have been constructed for disk and spherical geometry and have been widely used in explaining experimental results. On the other hand, there are two periodic geometries, torus and cylinder, which are useful for theoretical studies. There are several reasons why people care about these two periodic geometries. First, these geometries allow some freedom to tune the periodic boundary conditions and the geometry itself, making it convenient to calculate some topological quantities, such as Chern number and Hall viscosities. Second, the torus is the natural geometry to study Fermi sea states and crystal states since it can be mapped into a complex plane without defects. Third, the torus is the natural geometry to compare different topological states at the same filling factor because of the absence of "shift". Fourth, the cylinder is the natural geometry to study edge physics. It is also natural to view a cylinder as a quasi-one-dimensional system, which provides convenience to apply the density matrix renormalization group algorithm to the study of the fractional quantum Hall effect. Earlier, only several trial wave functions, such as the Laughlin wave function, the Moore-Read wave function, and the composite fermion Fermi sea wave function are known on a torus. In this thesis, we first construct the composite fermion wave functions for the general Jain states. We introduce a non-trivial projection method to construct wave functions in the lowest Landau level. We further show that the composite fermion wave functions we construct are very accurate descriptions of exact Coulomb eigenstates in the lowest Landau level. They allow the numerical study of large systems, which are not accessible in the exact diagonalization study in the torus geometry. We then apply these composite fermion wave functions to study Berry phase and Hall viscosities. In recent years, the issue of the nature of the composite fermion Fermi sea at $\nu=1/2$ has been of interest. In particular, the Berry phase associated with a loop around the Fermi surface is a criterion to determine whether a composite fermion is a Dirac Fermion. We have applied our lowest Landau level projection approach to the composite fermion Fermi sea to evaluate the Berry phase. We find the $\pi$ Berry phase for the projected composite fermion wave function, which other works have also reported. We further demonstrate that the Berry phase shifts away from $\pi$ with Landau level mixing. More importantly, the rate that the Berry phase rotates away from $\pi$ with the mixing of higher Landau level components increases with the system size. Hall viscosity is a geometric response of Hall fluid. It has been proposed as a topological quantity of quantum Hall fluid by Read. It can be evaluated by deforming the geometry of a torus. We evaluate Hall viscosities for a series of Jain states and showed they are consistent with Read's quantization relation. We show that with some assumption, the Hall viscosity can be derived analytically for the composite fermions and, more generally, the so-called "parton states". We also calculate Hall viscosities of various composite fermion Fermi seas at different fillings and find they cannot be viewed as naive limits of Jain states. Finally, we study the ``composite anyons" which have fractional statistics. They can be viewed as intermediate states between composite fermions when the numbers of vortices attached are fractions. We construct their wave functions on a torus, show the multi-component wave functions satisfy the braiding ground and have expected ground state degeneracy. We also use these anyon wave functions to calculate transport gaps, Chern numbers, and Hall viscosities. We also briefly introduce the composite fermion wave functions on a cylinder and composite fermion crystal wave functions on a torus in this thesis. We also give the outlook for future research at the end.