  The BTZ black hole belongs to a family of locally three-dimensional anti-de
Sitter (AdS$_3$) spacetimes labeled by their mass $M$ and angular momentum $J$.
The case $M \ell \geq |J|$, where $\ell$ is the anti-de Sitter radius, provides
the black hole. Extending the metric to other values of of $M$ and $J$ leads to
geometries with the same asymptotic behavior and global symmetries, but
containing a naked singularity at the origin. The case $M \ell \leq -|J|$
corresponds to spinning conical singularities that are reasonably well
understood. Here we examine the remaining case, that is $-|J|<M\ell<|J|$. These
naked singularities are mathematically acceptable solutions describing
classical spacetimes. They are obtained by identifications of the covering
pseudosphere in $\mathbb{R}^{2,2}$ and are free of closed timelike curves. Here
we study the causal structure and geodesics around these \textit{overspinning}
geometries. We present a review of the geodesics for the entire BTZ family. The
geodesic equations are completely integrated, and the solutions are expressed
in terms of elementary functions. Special attention is given to the
determination of circular geodesics, where new results are found. According to
the radial bounds, eight types of noncircular geodesics appear in the BTZ
spacetimes. For the case of overspinning naked singularity, null and spacelike
geodesics can reach infinity passing by a point nearest to the singularity,
others extend from the central singularity to infinity, and others still have a
radial upper bound and terminate at the singularity. Timelike geodesics cannot
reach infinity; they either loop around the singularity or fall into it. The
spatial projections of the geodesics (orbits) exhibit self-intersections, whose
number is determined for null and spacelike geodesics, and it is found a
special class of timelike geodesics whose spatial projections are closed.
