Let $(X_n)_{n}$ be a sequence of uniform spaces such that each space $X_n$ is
a closed subspace in $X_{n+1}$. We give an explicit description of the topology
and uniformity of the direct limit $u-lim X_n$ of the sequence $(X_n)$ in the
category of uniform spaces. This description implies that a function $f:u-lim
X_n\to Y$ to a uniform space $Y$ is continuous if for every $n$ the restriction
$f|X_n$ is continuous and regular at the subset $X_{n-1}$ in the sense that for
any entourages $U\in\U_Y$ and $V\in\U_X$ there is an entourage $V\in\U_X$ such
that for each point $x\in B(X_{n-1},V)$ there is a point $x'\in X_{n-1}$ with
$(x,x')\in V$ and $(f(x),f(x'))\in U$. Also we shall compare topologies of
direct limits in various categories.