This chapter explains the construction of a candidate for a fundamental solution and the existence, smoothness, and certain bounds for a fundamental solution Γ. The underlying assumptions were that ( a ij , ( x )) is uniformly positive definite and a ij , b i are bounded and uniformly Holder continuous. The chapter presents a proof of how if a ij , b i are Lipschitz continuous, uniformly in compact sets, then, for any Borel set A, P x ,( ξ(t) ∈A) = ∫ A Γ (x, t, y)dy , where Γ( x, t, y ) = Γ( x, t, y , 0). It presents a case where L is a degenerate elliptic operator. The concept of a fundamental solution Γ, if taken in the sense of the equation P x ,( ξ(t) ∈A) = ∫ A Γ (x, t, y)dy , still makes sense. The chapter explains interior and boundary estimates. It also explains estimates near infinity, behavior of ξ ( t ) near S.