  This paper focuses on curves and surfaces of constant width, with some
additional results about general ovals. We emphasize the use of Fourier series
to derive properties, some of which are known. Amongst other results, we show
that the perimeter of an oval is $\pi$ times its average width, and provide a
bound for the radius of curvature of an oval that depends on the structure of
the harmonics in its Fourier series. We prove that the density of a certain
packing of Reuleaux curved triangles in the plane is $$\frac{2 (\pi -
\sqrt{3})}{\sqrt{15} + \sqrt{7} - 2 \sqrt{3}} \simeq 0.92288$$ which exceeds
the maximum density for circles ($\simeq 0.9060$), and conjecture this is the
maximum for any curve of constant width. For surfaces of constant width we show
that $ \rho_0(P)+ \rho_1(Q)=w$ where the $\rho_i$ are the principal curvatures,
$P$ and $Q$ are opposite points, and $w>0$ is the width. Moreover, an ovoid is
a surface of constant width $w>0$ if and only if $\rho_{{\rm
mean}}(P)+\rho_{{\rm mean}}(Q) = w$ where $\rho_{{\rm mean}}(P)$ is the average
radius of curvature at point $P$. Finally, we provide a Fourier series-based
construction that produces arbitrarily many new surfaces of constant width.
