This dissertation studies the algebraic varieties arising from the conditional independence statements of Bayesian networks. Reduction techniques are described for relating these varieties to the varieties for smaller Bayesian networks. Particular attention is paid to the issues of primality, dimension, and degree. A classification of 5-node Bayesian networks is given based on whether or not they are prime for all state vectors. A proof of the Degree-2 Conjecture is given for a subclass of Bayesian networks which includes those with binomial global Markov ideal.