Linear multistep methods are used for the numerical solution of ordinary differential equations. Conceptually, a numerical method starts from an initial point and then takes a short step forward in time to find the next solution point. The process continues with subsequent steps to map out the solution. Single-step methods (such as Euler's method) refer to only one previous point and its derivative to determine the current value. Methods such as Runge–Kutta take some intermediate steps (for example, a half-step) to obtain a higher order method, but then discard all previous information before taking a second step. Multistep methods attempt to gain efficiency by keeping and using the information from previous steps rather than discarding it. Consequently, multistep methods refer to several previous points and derivative values. In the case of linear multistep methods, a linear combination of the previous points and derivative values is used.

Numerical methods for ordinary differential equations approximate solutions to initial value problems of the form $${\displaystyle y'=f(t,y),\quad y(t_{0})=y_{0}.}$$

The result is approximations for the value of ${\displaystyle y(t)}$ at discrete times ${\displaystyle t_{i}}$: $${\displaystyle y_{i}\approx y(t_{i})\quad {\text{where}}\quad t_{i}=t_{0}+ih,}$$ where ${\displaystyle h}$ is the time step (sometimes referred to as ${\displaystyle \Delta t}$) and ${\displaystyle i}$ is an integer.

Multistep methods use information from the previous ${\displaystyle s}$ steps to calculate the next value. In particular, a linear multistep method uses a linear combination of ${\displaystyle y_{i}}$ and ${\displaystyle f(t_{i},y_{i})}$ to calculate the value of ${\displaystyle y}$ for the desired current step. Thus, a linear multistep method is a method of the form $${\displaystyle {\begin{aligned}&y_{n+s}+a_{s-1}\cdot y_{n+s-1}+a_{s-2}\cdot y_{n+s-2}+\cdots +a_{0}\cdot y_{n}\\&\qquad {}=h\cdot \left(b_{s}\cdot f(t_{n+s},y_{n+s})+b_{s-1}\cdot f(t_{n+s-1},y_{n+s-1})+\cdots +b_{0}\cdot f(t_{n},y_{n})\right)\\&\Leftrightarrow \sum _{j=0}^{s}a_{j}y_{n+j}=h\sum _{j=0}^{s}b_{j}f(t_{n+j},y_{n+j}),\end{aligned}}}$$ with ${\displaystyle a_{s}=1}$. The coefficients ${\displaystyle a_{0},\dotsc ,a_{s-1}}$ and ${\displaystyle b_{0},\dotsc ,b_{s}}$ determine the method. The designer of the method chooses the coefficients, balancing the need to get a good approximation to the true solution against the desire to get a method that is easy to apply. Often, many coefficients are zero to simplify the method.

One can distinguish between explicit and implicit methods. If ${\displaystyle b_{s}=0}$, then the method is called "explicit", since the formula can directly compute ${\displaystyle y_{n+s}}$. If ${\displaystyle b_{s}\neq 0}$ then the method is called "implicit", since the value of ${\displaystyle y_{n+s}}$ depends on the value of ${\displaystyle f(t_{n+s},y_{n+s})}$, and the equation must be solved for ${\displaystyle y_{n+s}}$. Iterative methods such as Newton's method are often used to solve the implicit formula.

Sometimes an explicit multistep method is used to "predict" the value of ${\displaystyle y_{n+s}}$. That value is then used in an implicit formula to "correct" the value. The result is a predictor–corrector method.

Consider for an example the problem $${\displaystyle y'=f(t,y)=y,\quad y(0)=1.}$$ The exact solution is ${\displaystyle y(t)=e^{t}}$.

A simple numerical method is Euler's method: $${\displaystyle y_{n+1}=y_{n}+hf(t_{n},y_{n}).}$$ Euler's method can be viewed as an explicit multistep method for the degenerate case of one step.