  We establish new bounds on the number of tangencies and orthogonal
intersections determined by an arrangement of curves. First, given a set of $n$
algebraic plane curves, we show that there are $O(n^{3/2})$ points where two or
more curves are tangent. In particular, if no three curves are mutually tangent
at a common point, then there are $O(n^{3/2})$ curve-curve tangencies. Second,
given a family of algebraic plane curves and a set of $n$ curves from this
family, we show that either there are $O(n^{3/2})$ points where two or more
curves are orthogonal, or the family of curves has certain special properties.
  We obtain these bounds by transforming the arrangement of plane curves into
an arrangement of space curves so that tangency (or orthogonality) of the
original plane curves corresponds to intersection of space curves. We then
bound the number of intersections of the corresponding space curves. For the
case of curve-curve tangency, we use a polynomial method technique that is
reminiscent of Guth and Katz's proof of the joints theorem. For the case of
orthogonal curve intersections, we employ a bound of Guth and the third author
to control the number of two-rich points in space curve arrangements.
