  In various subjects including mathematics, one can hope to use mathematical
thinking well when the right kinds of algebraic structure to consider can be
discovered or spotted. Therefore, it would help to understand kinds of
algebraic structure in some great generality. In tradition, certain general
kinds of algebraic structure are studied through the theory of operads, of
algebraic theories, of properads (possibly with "colours" or "sorts") or of the
like. We understand this as use of algebra for studying a certain meta aspect
of the subject of algebra, namely, studying kinds of algebraic structure in
general (rather than structures of specific kinds themselves).
  In higher categorical contexts, more various algebraic structures can be
considered (starting in fact with operads considered with arbitrarily varying
colours) than can be covered with mere higher categorified versions of the
traditional tools. In this survey, we develop a systematic view on quite
general algebraic structures with high categorical dimensionality. We do this
by extending algebra for the meta aspect of algebra, to a higher order, or
highly meta, theory of algebra, where the theoretic order turns out to match
the categorical dimension. We will discuss some examples including a natural
generalization of the notion of topological field theory (TFT) obtained in the
resulting new system. It turns out a generalized TFT can have a very different
characterization from a conventional TFT does.
