The mathematics of elliptic curves has been studied for a decade in the fields of number theory and arithmetic geometry. The use of elliptic curves in cryptography was suggested independently by Koblitz and Miller in 1985. Since then, many scientists have worked on elliptic curve cryptography and have studied several subjects of this area of research. Now a days, the main important application of elliptic curves is in cryptography. The traditional model to define an elliptic curve is the so called Weierstrass equation. Over the last 30 years many scientists have evaluated different forms of elliptic curves, also have introduced several new forms and coordinate systems to improve the efficiency of elliptic curve cryptography. In 2007 , Edwards introduced a new normal form for elliptic curves and presented its addition low. Every elliptic curve over a non-binary field can be converted to a curve in Edwards form over an extension of the ground field. After the presentation of Edwards, Bernstein and Lange observed the impact of the proposed form in elliptic curve cryptography. They extended this form and provided the addition and doubling formulas improving the efficiency and speed. The addition formulas are unified, i.e., work for doubling. Moreover, for a subfamily of elliptic curves, the addition formulas have the feature of completeness, i.e., work for any pair of points without exception. The proposed form is known as the "Edwards" form. Latter on, Bernstein et.al. suggested the inverted Edwards coordinates to improve the speed of the point addition computation. Furthermore, they extended the family of Edwards to the family of twisted Edwards curves to cover more isomorphism normal; MARGIN: 0cm 0cm 0pt; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt" elliptic curves. In other words, the family of twisted Edwards curves over a finite field is the family of elliptic curves with a subgroup of order 4. Recently, there has been a fast improvement of Edwards curves and their generalization to apply in cryptography. Currently, the twisted Edwards curves provide the fastest addition formulas among all other forms of elliptic curves over finite fields in odd characteristic. Therefore, using Edwards curves in cryptography is recommended. In this thesis, we study the families of Edwards curves and their extensions. We describe the addition and doubling formulas in different families of Edwards curves. We explain the properties of Edwards curves and investigate their efficiency to use in elliptic curve cryptography.