  In this thesis, we consider semi-algebraic sets over a real closed field $R$
defined by quadratic polynomials. Semi-algebraic sets of $R^k$ are defined as
the smallest family of sets in $R^k$ that contains the algebraic sets as well
as the sets defined by polynomial inequalities, and which is also closed under
the boolean operations (complementation, finite unions and finite
intersections). We prove new bounds on the Betti numbers as well as on the
number of different stable homotopy types of certain fibers of semi-algebraic
sets over a real closed field $R$ defined by quadratic polynomials, in terms of
the parameters of the system of polynomials defining them, which improve the
known results. We conclude the thesis with presenting two new algorithms along
with their implementations. The first algorithm computes the number of
connected components and the first Betti number of a semi-algebraic set defined
by compact objects in $\mathbb{R}^k$ which are simply connected. This algorithm
improves the well-know method using a triangulation of the semi-algebraic set.
Moreover, the algorithm has been efficiently implemented which was not possible
before. The second algorithm computes efficiently the real intersection of
three quadratic surfaces in $\mathbb{R}^3$ using a semi-numerical approach.
