The braid groups were mainly studied by E. Artin in 1925. After that a lot of improvement has been seen in this area and the theory was established as a theory of braids. For example J. S. Birman and P. Dehory, etc, have great contributions in this subject. This book contains the basic tions of braids and braid moids. Using the Bokut's n-commutative Grobner bases we have given an algorithm to compute inductively the Hilbert series of all the braid moids and we gave examples for few initial values. Here we have given the growth rates of braid moids with three and with four strings and shown that the growth functions of these moids are exponential. In this book we have described the Garside elements of some spherical Artin moids. At the end we have proved that for the spherical Artin moids the growth rate is bounded above by 4.