The paper is concerned with Morse theory for asymptotically quadratic functionals on a Hilbert space

$X$. Let

${x}_{0}$ be an isolated critical point and

$f\in {C}^{2}(X,\mathbf{R})$. Then

${x}_{0}$ may be classified by its critical groups

${c}_{k}(f,{x}_{0}):={H}_{k}(\{f\le c\},\{f\le c\}\setminus \left\{{x}_{0}\right\})$, where

$c=f\left({x}_{0}\right)$. In this paper the notion of critical group at infinity is introduced and it is shown that such groups have similar properties to those at finite points. In particular, if

${f}^{\text{'}\text{'}}\left(\infty \right)$ is nondegenerate and has Morse index

$\mu $, then

${c}_{k}(f,\infty )\ne 0$ if and only if

$k=\mu $. It is known that if

${f}^{\text{'}\text{'}}\left({x}_{0}\right)$ is degenerate and

$f$ satisfies the so-called local linking condition at

${x}_{0}$, then

${c}_{k}(f,{x}_{0})\ne 0$ for some

$k$. A similar result is shown to hold for

${c}_{k}(f,\infty )$. Moreover, a new “angle condition” is introduced under which the

${c}_{k}$’s behave as in the case of nondegenerate critical point. Applications of the above results are given to the Dirichlet problem

$-{\Delta}u=p(x,u)$ in

${\Omega}$,

$u=0$ on

$\partial {\Omega}$, where

${\Omega}$ is a bounded domain in

${\mathbf{R}}^{N}$,

$p(x,0)=0$ and

$p$ is asymptotically linear. It is shown that under different conditions

${c}_{k}(f,0)\ne {c}_{k}(f,\infty )$ for some

$k$, and therefore this problem has a nontrivial solution. Particular attention is paid to the resonant case.