The main goal of this thesis is the study of elliptic curves over finite fields and the number of points of them. A study of interest is to analyze and find the cases when the difference #E(F_q^n )-#E(F_q) vanishes, where #E(F_q) denotes the number of points of an elliptic curve E over the finite field F_q. Moreover we can show that the sequence a_n=#E(F_q^n ) - #E(F_q)=q^n-q for n odd, q prime of the form q=4k+3 where k is a nonnegative integer and any elliptic curve of the form E : y^2=x^3+tx over F_q. Also a_n=#E(F_q^n ) - #E(F_q)=q^n-q for q prime of the form 3k+2 where k is a positive integer, n odd and any elliptic curve of the form E: y^2=x^3+b over F_q.