A smooth curve $\gamma: [0,1] \to \Ss^2$ is locally convex if its geodesic curvature is positive at every point. J.~A.~Little showed that the space of all locally positive curves $\gamma$ with $\gamma(0) = \gamma(1) = e_1$ and $\gamma'(0) = \gamma'(1) = e_2$ has three connected components $\cL_{-1,c}$, $\cL_{+1}$, $\cL_{-1,n}$. These spaces and variants have been discussed, among others, by B.~Shapiro, M.~Shapiro and B.~Khesin. Our first aim is to describe the homotopy type of these spaces. The connected component $\cL_{-1,c}$ is known to be contractible. We construct maps from $\cL_{+1}$ and $\cL_{-1,n}$ to $\Omega\Ss^3 \vee \Ss^2 \vee \Ss^6 \vee \Ss^{10} \vee \cdots$ and $\Omega\Ss^3 \vee \Ss^4 \vee \Ss^8 \vee \Ss^{12} \vee \cdots$, respectively, and show that they are (weak) homotopy equivalences. More generally, a smooth curve $\gamma: [0,1] \to \Ss^n \subset \RR^{n+1}$ is locally convex if $\det(\gamma(t),\ldots,\gamma^{n}(t)) > 0$ (for all $t$). We would like to understand the homotopy type of the space $\cL$ of locally convex curves with $\gamma^{(j)}(0) = \gamma^{(j)}(1) = e_{j+1}$ (for all $j \ne n$). We describe a CW complex with the same homotopy type. The homotopy type of $\cL$ is described for $n = 3$.