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This paper has two parts. We first survey recent efforts on the Bloomconjecture which still remains open in the case of complex dimension at least4. Bloom's conjecture concerns the equivalence of three regular types. There isa more general important notion, called the singular D'Angelo type (or simply,D'Angelo type). While the finite D'Angelo type condition is the right one forthe study of local subelliptic estimates for Kohn's$\overline{\partial}$-Neumann problem, regular types are important as theirfiniteness gives the global regularity up to the boundary of solutions ofKohn's $\overline{\partial}$-Neumann problem.  In the second part of the paper, we provide a proof of a seemingly elementarybut a truly fundamental property (Theorem 2.2 or its CR version Theorem 2.5) onthe vanishing order of smooth functions along a system of non-integrable vectorfields. A special case, Corollary 2.6, of Theorem 2.5 had already appeared in apaper of D'Angelo. A main goal in this part is to provide proofs for theseresults for the purpose of future references. Our arguments are based on a deepnormalization theorem for a system of non-integrable vector fields due toHelffer-Nourrigat, as well as its late generalization in Boauendi-Rothschild.

Regular types and order of vanishing along a set of non-integrable vector fields