this list, although impressive, does not contain all interesting systems. This motivates the search for other properties, call them *specification-like*, which may be used to examine systems without specification in Bowen’s sense. In this survey we describe various notions designed to replace specification. It turns out that there are many systems lacking the specification property, but exhibiting a weaker version of it, which suffices to derive interesting results. This approach has been used to study systems with some forms of non-uniform hyperbolicity, such as $\beta$-shifts.

![The connections between various generalizations of the specification property. There are no more implications between these notions besides those following by transitivity.[]{data-label="fig:rys"}](rysunek2){width="\textwidth"}

The length of this paper does not allow detailed exposition of all aspects of the theory of specification-like properties. We would like to concentrate on the “big picture”, presenting a broad overview of possible generalizations of the specification property and discussing various examples illustrating dynamical systems with these properties. Figure \[fig:rys\] presents a diagram summarizing the specification-like properties we discuss. We also describe examples illustrating the fact that none of the implications presented on Figure \[fig:rys\] can be reversed. Some of them have never been published before. We would like to add to 