  We study the problem of partitioning a given simple polygon $P$ into a
minimum number of polygonal pieces, each of which has bounded size. We give
algorithms for seven notions of `bounded size,' namely that each piece has
bounded area, perimeter, straight-line diameter, geodesic diameter, or that
each piece must be contained in a unit disk, an axis-aligned unit square or an
arbitrarily rotated unit square.
  A more general version of the area problem has already been studied. Here we
are, in addition to $P$, given positive real values $a_1,\ldots,a_k$ such that
the sum $\sum_{i=1}^k a_i$ equals the area of $P$. The goal is to partition $P$
into exactly $k$ pieces $Q_1,\ldots,Q_k$ such that the area of $Q_i$ is $a_i$.
Such a partition always exists, and an algorithm with running time $O(nk)$ has
previously been described, where $n$ is the number of corners of $P$. We give
an algorithm with optimal running time $O(n+k)$. For polygons with holes, we
get running time $O(n\log n+k)$.
  For the other problems, it seems out of reach to compute optimal partitions
for simple polygons; for most of them, even in extremely restricted cases such
as when $P$ is a square. We therefore develop $O(1)$-approximation algorithms
for these problems, which means that the number of pieces in the produced
partition is at most a constant factor larger than the cardinality of a minimum
partition. Existing algorithms do not allow Steiner points, which means that
all corners of the produced pieces must also be corners of $P$. This has the
disappointing consequence that a partition does often not exist, whereas our
algorithms always produce useful partitions. Furthermore, an optimal partition
without Steiner points may require $\Omega(n)$ pieces for polygons where a
partition consisting of just $2$ pieces exists when Steiner points are allowed.
