This paper deals with tilings, called similarity tilings, of the plane by tiles with (up to similarity) only finitely many shapes and finitely many surroundings (or matching conditions). It is called inflationary if there is an expanding linear map which maps tiles onto unions of tiles. The main result is the following theorem: A complex number \(\gamma\), \(| \gamma | > 1\), is the expansion factor for an inflationary similarity-tiling of the plane (both by general tiles, and by polygonal tiles) if and only if \(\gamma\) is algebraic. Moreover, for every such \(\gamma\) there is also an inflationary similarity-tiling by polygons. (In contrast to this, for tilings of the real line, any real number occurs as an expansion factor.) The proof is based on an approach (due to W. P. Thurston) which is presented in some detail. Quasiperiodic and inflationary tilings are characterized in the space of (equivalence classes of) similarity-tilings furnished with a suitable topology, and are closely related to Markov partitions of (smooth hyperbolic) dynamical systems. The construction of suitable inflationary tilings uses Pisot numbers.