In 1924, P. Alexandroff proved that a locally compact $T_2$-space has a $T_2$ one-point compactification (obtained by adding a ``point at infinity'') if and only if it is non-compact. He also asked for characterizations of spaces which have one-point connectifications. Here, we consider one-point connectifications, and in analogy with Alexandroff's theorem, we show that in the realm of $T_i$-spaces ($i=3\frac{1}{2},4,5$) a locally connected space has a one-point connectification if and only if it has no compact component. We explain how this theorem may be extended to the case $i=2$ and to the case $i=6$ under the set-theoretic assumption $\mathbf{MA}+\neg\mathbf{CH}$. Contrary to the case of the one-point compactification, a one-point connectification, if exists, may not be unique. We consider the collection of all one-point connectifications of a locally connected locally compact space in the realm of $T_i$-spaces ($i=3\frac{1}{2},4,5$). This collection, naturally partially ordered, is a compact conditionally complete lattice whose order-structure is rich enough to determine the topology of all Stone-Čech remainders of components of the space.