In this thesis we will look at methods for constructing elliptic curves over Q with high ranks. Using these methods we find an elliptic curve with rank at least 13, an infinite family of elliptic curves with rank at least 9, an elliptic curve with rank at least 10 and torsion-subgroup Z/2Z and an infinite family of elliptic curves with rank at least 8 and with torsion point of order 2.